Finite groups with small God's numbers Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word of length $\le N(S,G)$. A famous example is the group of Rubik's Cube where the answer is known to be 26 (if $S$ is the set of all quarter twists) and this is known as God's number. 
Where can I find more example of this kind, i.e. groups where $|S|$ and $N(S,G)$ are know (or there is at least some bound) and we have that $|S|,N(S,G)\ll |G|$?
More generally, can we expect that for most groups we can find some generating set $S$ such that $|S|,N(S,G)\ll |G|$?
 A: This is one of the fundamental questions in the theory of groups known as the width problem.
One of the best (general) references devoted to this problem is the book Words: Notes on Verbal Width in Groups by Dan Segal.
Yet another good reference is the paper Width questions for finite simple groups by Martin Liebeck.
A well-known conjecture due to Babai states that there must exists a constant $c$ such that $N(S,G)<(\log|G|)^c$ for all non-abelian finite simple groups $G$ and generating sets $S$.
L. Babai and A. Seress, On the diameter of permutation groups, European J. Combin. 13 (1992), 231–243.
A: The requirement $|S|\ll |G|$ looks odd to me as $S$ is a subset of $G$. As to $N(S,G)$ -- well, at least for finite abelian groups it is known to be strictly smaller than the size of the group. In fact, if $G$ is finite abelian of the type $(m_1,\dotsc, m_r)$ with $m_1\mid\dotsb\mid m_r$, then the largest possible value of your $N(S,G)$ as $S$ varies over all generating sets is $(m_1-1)+\dotsb+(m_r-1)$.
