The Group Oh

There are four cubes, which exactly have all possible symmetries of the cube, one of them - the Superflip - needs 20 moves to be generated. Historically this was the first cube which was proven to need 20 moves, and this is still the best lower bound for the diameter of the cube group.

The next quite trivial table gives the number of cubes up to M-symmetry which exactly have Oh-symmetry, the average maneuver length is 15:

Distance
Number
Distance
Number
0f
1
19f
1
6f
1
20f
1

The maneuver for the Superflip also is included in the file 20moves.zip.
Name
 Identitiy
Generator
 (0f*)
Name
 Pons Asinorum
Generator
 B2 F2 L2 R2 D2 U2 (6f*)
Name
 Pons Asinorum composed with Superflip
Generator
 R B L F D L R' B2 F D' U L D' U R' D' F' R' U' (19f*)
Name
 Superflip
Generator
 D' R2 F' D2 F2 U2 L' R D' R2 B F R' U2 L' F2 R' U2 R' U' (20f*)

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© 2017  Herbert Kociemba