I hate the G perm the most, I can prove it.

Note: This a rather cube-oriented post, and will be better if you have a cube with you, and will be a lot better if you're fully familiar with Singmaster notation.

Warning: These algorithms are only useful for academic purposes and will probably just slow you down in the long run. Learn the lesson, not the algorithms.

I still don't do CFOP that well, and if you're just learning it now, you should have fewer bad habits than me because you have the benefit of years of refinement of the method even before you ever saw any algorithms.

I was rather skeptical about the CFOP method when I first encountered it, because I didn't think that most people would be able to memorize algorithms for 80ish last layer cases (57 orientation cases, or OLLs and 21 permutation cases, or PLLs) in addition to some specialized F2L (first two layer) cases. But, I was using a method that needs less than 20 algorithms and had just barely caught up to the speed of its creator only years later. In the meantime, CFOP times kept going down. I later figured out that there were potentially a decent number of mirror image cases that kept the number of cases manageable.

I also realized that when I had used a layer by layer method, I had the decided disadvantage of having learned one (Nourse) that left the solved layer up instead of one that leaves the solved layer down (Taylor). While I saw the move efficiency of putting the middle layer edges in with the first layer corners, I still had been working with the bad habit of not turning the cube over until I finished the first two layers.

When I finally accepted that I should start learning some PLL's, the two major resources I was looking at was speedcubing.com, and Jessica Fridrich's page. I probably started with the U perm, because I figured out that R2US'U2SUR2  is just the same as the U2M'U2M edge 3-cycle with some setup moves. I was OK with left-handed and right-handed versions of that easily enough. I may have learned the T perm next, and was so glad that it was symmetrical and I didn't have to learn a second version of it. Strangely, the version of the T perm that I learned is the inverse of one of the algorithms on the speedcubing.com page (but hey - it's symmetrical so it's also functionally its own inverse), so I'm not sure how I got to that version of the algorithm. But, I digress - and it's going to look like I'm doing it some more but bear with me. If you want to play along, this would be a good time to grab a cube.

For the N perm, I took the easy-to-memorize route, and went with  

LU'RU2L'UR' LU'RU2L'UR' U 

because the first segment and the second segment are the same thing. This was the first of the PLLs that I learned that had a diagonal swap, do I had to use this a lot when I was still doing 4 look last layer.

So if you start with a solved cube, and do the first segment of that move, you get the fr and bl F2L pairs swapped, and you get two U-layer corner-edge pairs swapped, like the J perm. Doing the second segment resolves the two F2L pairs, and swaps the second set of U layer corner-edge pars, creating the N perm.

I also figured out that if you inserted a U' or a U move in the middle of the N perm above, cancelling part of the U layer swap from the first segment to the second, you got a J perm. I later learned a better algorithm for one of the J perms, but I still use the method above for the other.

So, in my weird bag of cube tricks, I knew that there was another way to fix the 2 F2L pairs that were undone by the first part of the N perm. Start with a solved cube again.

LU'RU2L'UR'  y  L'R'U2LR gets you something that looks like... Could it be? The G perm! So now, I just had to reverse engineer it. (Put down your cube momentarily.)

The G perm isn't really all the way symmetrical the way we would like, and it's not its own inverse. Therefore, you need four versions of the move. Left-handed, right-handed, left inverse, and right inverse.

The version of the G perm that the above example solves is with headlights on the front, and a bar on the back with the ub and the ubr pieces. The mirror image is still headlights on the front and the bar between ub and ubl. For a long while, this was as far as I was, as I had not bothered to write down the move and properly figure out the inverse.

Headlights on the back, bar on the right between ur and ufr, which is what you might have if you did the above bold move from a solved cube, (OK pick up the cube again...) is solved with

L'R'U2LR y' LU'RU2L'UR'.

And you should be back to a solved cube.

Now you could try the other two, the mirror images. Like before, one will undo the other.

R'UL'U2RU'L y' RLU2L'R', 

and LRU2L'R' y R'UL'U2RU'L.

So, while I can't recommend these algorithms to anyone other than just for academic interest or just basic insight into the algorithm building process, I hope that someone can use this to actually make some better algorithms.