Burnside's lemma is a clever tool to calculate the number of distinct configurations up to a given symmetry group.
Wikipedia's example shows that the number of rotationally distinct colourings of the faces of a cube using three colours is 57. This introduction similarly shows that there are 6 distinct squares whose vertices have one of two colours.
But what if we actually want to construct the 57 distinct-coloured cubes, or the 6 distinct-coloured squares, instead of just counting them? Is there a non-brute-force way of doing this?