30 different painted cubes.
Let the six faces be a, b, c, d, e, and f. With face a opposite face b there are six arrangements for the other four colours around the cube: cdef, cdfe, cedf, cefd, cfde and cfed. Likewise for the face a opposite face c ; face a opposite face d ; face a opposite face e ; and face a opposite face f. All have six arrangements for the remaining five colours. Hence the total is 5 x 6 = 30 arrangements.