(a) I would like to know whether there is a group theoretic approach for calculating the diameter of the Cayley graph of Rubik's Cube group.
I know it's been proved that the above diameter is $20$ but the approach uses brute force.
Also I wonder whether there is a "nice" presentation of this group (it is a finite group so the relations between the elements of this group come from the multiplication table, but (b) is there a systematic way of writing down these relations?)
What about a $2\times2\times2$ Rubik's Cube? (Is it still hard to examine the Cayley graph?)
EDIT
Since there has been no full answer to my initial questions I'd like to ask for something else too:
Is there an "nice" way to describe a maximal tree of the Cayley graph of the $3\times3\times3$ Rubik's Cube?