I considered this question in teaching abstract algebra. I am trying to impress upon my students about the messiness of studying nilpotent groups generally. More specifically, I am looking for ways of constructing families of $2$-groups which demonstrate this in a fairly transparent and somewhat transcendental manner.
For example, is there some construction which uniquely yields a non-trivial $2$-group for every graph (likely with some special properties); this is what I mean by "transparent". When I say "transcendental", I mean the that the idea should transcend the usual things one sees in typical graduate sequence textbook in abstract algebra. It is easy, for example, to form groups of unipotent matrices over fields of characteristic $2$, but I am looking for something that displays more chaos and disorder. Graphs seem like a good place to start because they are so intuitive.
I hope this question is not too vague. I would personally like to better understand $p$-groups generally, so I am asking this with a somewhat selfish motive. Of course, it would be nice to know something that my students would be able to grasp. Thank you.
If you take a finite rooted binary tree then its automotphism group is a finite $2$-group. For infinite $2$-groups consider the Grigorchuk group. It is easy to define but pretty transcendental.