is it 6^6/24 . If you write the cycle index of group of permutation of faces of cube this is what you get.
Since each face have a unique color, ans would be 6!/24.Open up the cube. Now, a cube can be opened in 24 ways(i.e) 6 ways to fix the bottom face and 4 ways of then opening it, thereby 24 ways. Each of the face can be colored in 6! ways but many cases would repeat due to opening of the cube.Thus, total possible coloring combinations = 6!/24 = 30