It is clear that the group G of the cube (the one with
4.3252x10^19 elements) can be embedded in a
symmetrical group, e.g. S_48, since each move of the cube can be
seen as a permutation of 48 objects. Hence, there is a smallest
number n such that G can be embedded in S_n. I'm curious to find
out what this number is.

It can be shown with some counting arguments that n>=32 (I'm
happy to write these down but it's nicer if you thought about
this first). I would be surprised if n=32 but you never know.
--
Michiel Boland <boland@sci.kun.nl>
University of Nijmegen
The Netherlands