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

My permutation group memory is rusty. Is it the "degree" of G you're
asking for?

I also get n = 32 as the smallest |S_n| divisible by |G|. I suspect
one can argue |G| must divide |A_n| (still requiring only n >= 32),
and if we can argue |G|*12 must divide |S_n| (accounting for all
parities of edges and corners) we get n >= 33.

On the flip side, I'll assert G has degree <= 42 (which is less than
the obvious representation in S_48). If anyone can prove me either
right or wrong, please do so. My assertion is based on the following
hand-waving argument:

G is a wreath product (or some kind of product) of the following
subgroups:
A:  S_8   8! corner positions
B:  3^7   3^7 corner orientations
C:  A_12  12!/2 edge positions
D:  2^11  2^11 edge orientation

Clearly subgroup A has degree 8 and C has degree 12.

I claim (wave hands wildly:-) that BxD has degree at most 22, since it
can be embedded in an S_22. I use every even factor of 22! for a component
of 2^11, and every third factor of 22! for a component of 3^7.

Divisibility arguments suggest some smaller S_n might embed 2^11x3^7,
but I don't know whether such embeddings can be realized. This is an
obvious area to consider for lowering the upper bound on degree(G). By
divisibility we can conceivably embed 2^11x3^7 in an S_18, but not in
an S_17.

Finally, the degree of a wreath(?) product is bounded by the sum of
the degrees of the multiplicands, hence
degree(G) <= 8 + 12 + 22 = 42.

If this (alleged) degree 42 construction is valid, is 42 minimal? I
strongly doubt it.