Technically the symmetry group of the rubiks cube is the symmetry group of the cube with all its label peeled off. The normal rubiks cube with all its faces painted different colors has trivial symmetry group because nothing preserves it.
My goal is to find a geometric realization of $Q_8$ and for some reason unknown to me I think it might occur as the restriction of the symmetries of either a $2\times 2\times 2$ cube suitably painted (for example maybe the top and bottom are both the same color, then there is a chessboard type coloring around the sides) or maybe an elongated $2\times 2$ (similar to how to Klein 4 group arises).
I've found GAP code that claims to give the symmetry group of the peeled $2\times 2\times 2$ http://cubeman.org/2x2x2.txt
How could I find the symmetry group of a painted cube?