Let's say we have to build a cube of shape N x N x N (in some units) by stacking smaller cubes each of size 1 x 1 x 1 (in some units). Each of the smaller cubes (unit cubes) come in one of the K distinct colors and we have sufficient supply of cubes of each color. How many distinct ways are there in which we can stack these cubes to form the bigger cube?
Bigger cubes which can be rotated or mirrored to each other are not considered distinct.
PS: I came across several links on the stackexchange but none of them are general enough. Very curious to know if there is a general formulation which can help solve this problem. Also interested in knowing what happens when we increase the dimensionality in the problem i.e. for N x N x N x N hypercube.
What you should know is the automorphism group of the cube, plus Burnside's lemma.
See if you can figure it out by assembling information from these references.