How many ways are there to put 8 beads of different colors on the vertices of a cube if rotations of the cube but not reflections are considered the same?

The answer is 1680.

When the cube is rotated, there are 8 points a single vertex could end up on, and 3 ways for each vertex to be rotated onto itself.

Therefore, it is 8!/(8x3)=40320/24=1680