if the above group is correct, then next we have to consider the kind of actions that preserve a solved cube's state visually (as in, the resulting state also looks solved):



permutations of centers that do not exchange faces (there are 4!6 except we're restricted to only positive ones, so 4!6/2)


whole cube rotations (4!)

(exchanging a pair of edge pieces is never possible, because they would end up rotated... so, different state)

then the set of visually distinct reachable states is the quotient set of the puzzle's group by the above (generated) subgroup.

the order of the whole group is |G| = 37 8! (24! 24!)/2, and the two subgroups above have trivial intersection, so the cardinality of the state set turns out to be |G| / (4!6/2 * 4!) ≈ 7.4e45. a random internet paper reaches the same result