By Igor Kriz and Paul Siegel, at Scientific American: Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups".
The Rubik's cube can be said to be a physical embodiment of the Rubik group, which is a certain subgroup of S48. (Why 48? There are 54 "facets" of the Rubik's cube, nine on each side; six of these don't move, leaving 48.) This subgroup has an easy presentation with six generators, namely rotations of the six faces. As the authors point out, other "Rubik's-type" puzzles embody groups of the same general sort.
The authors have invented puzzles based on certain sporadic groups, namely theMathieu groups M12 and M24 and the Conway group Co0. These right now only exist as computer programs, although the authors claim that a physical version of the M24 puzzle could be built.
A possibly interesting, although not-at-all well-defined question -- which groups are "buildable", in the sense that one can build a physical object that represents them?
The programs (which run on Windows machines) can be downloaded from Igor Kriz's home page.
Thanks to John Armstrong for pointing me to the article.