1.61 Using 6 different colors to paint the 6 faces of a cube, so that different faces are painted with different colors, how many different painting methods are there (painting methods that are still different after any rotation of the cube are considered different)?
[Solution] To paint the 6 faces of a cube with 6 different colors, there are $6!=720$ ways of painting. However, some of these painting methods are merely different in position; they can become the same by rotating the cube. According to the problem, such two painting methods are considered the same.

For a cube that has already been painted with 6 different colors, one of the 6 faces can be chosen as the bottom face, giving 6 different choices. Then, one of the 4 side faces can be chosen as the front face, giving 4 different choices. Thus, this cube has 24 different positions, which means in the above counting, each painting method is counted 24 times. Therefore, the number of different painting methods is 30.