Incomplete open cubes

How many ways can a cube be incomplete? What I mean by that is, if you take away some of its edges, which leaves kind of a partial frame thing like this, how many distinct configurations can you get? And the tricky part here is that we're going to consider incomplete cubes to be the same if you can rotate one to match the other. Now, even though this is a very pure math puzzle, if you add in a few other constraints on the cubes. This was essentially the premise behind a work of modern art by the artist Sol LeWitt in 1974. I've been away from making videos this summer, and in that time I've been using Patreon funds to commission a set of guest videos, and the most recent one tells the story behind this artwork in parallel with the story of a problem solving process that could lead you to answer this counting question. And effectively, that process is to rediscover a bit of group theory and a very beautiful fact within it known as Burnside's Lemma. The creator for this video is a friend of mine, Paul Dancstep, and I thought he did a really beautiful job creating something that could be at once accessible to a middle schooler and thought provoking to a PhD student. So I wanted to give it a very brief shout out here in the shorts feed for next time you find yourself in the mood to sit down for a longer piece. In the meantime, I thought you'd enjoy starting to think about the puzzle itself.