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Rubik's cube group

The Rubik's Cube group is a mathematical group which corresponds to the set of all cube operations on Rubik's Cube, with function composition (chaining moves) as the group operation.

Any set of operations which returns the cube to the solved state, from the solved state, should be thought of as the identity transformation (the operation that does nothing). Any set of operations which solves the cube from a scrambled state should be thought of as an inverse transformation of the given scrambled state, since it returns the identity transformation.

Total number

The order of the cube group G is then equal to the number of possible positions attainable by the cube. This is:

|G| = 1/12 8! 3^8 12! 2^12 = 43,252,003,274,489,856,000

which factorizes as:

|G| = 2^27 3^14 5^3 7^2 11^1

Because of the large size of the cube group, it is sometimes useful to analyse the structure with the assistance of a computer algebra system.

Source: Wikipedia - Rubik's cube group