The group G of all legal cube states.
Sexy Move = [R, U] = RUR−1U−1. The basic commutator — it has order 6 and only displaces 8 pieces.
Sune-like = F[R, U]F−1. The commutator [R, U] conjugated by F. Conjugation preserves cycle structure so this also has order 6.
Sune = RUR−1URU2R−1 = UR · U · (U2)R. A product of three conjugates of powers of U by R (where the middle U is conjugation by R0 = e). It has order 6 and permutes only 3 corners.
Commutators: Many solving algorithms are based on commutators [A, B] = ABA-1B-1. These allow you to affect only a small number of pieces while leaving the rest of the cube unchanged.
Order of G: The number of possible positions is 43,252,003,274,489,856,000 (approx. 43 quintillion).
Generators: The group is generated by the 6 face rotations: {U, D, L, R, F, B}. Each generator has order 4.
Structure: The Rubik's group is a subgroup of the permutation group S48. It is not simple; it has a large normal subgroup consisting of positions where the pieces are in their home locations but rotated/flipped.
God's Number: Every position can be solved in 20 moves or fewer in the half-turn metric.