> Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S_n$?

Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations? 
This looks like a basic question, but I seem not to be able to find answers or even this question in the literature. If this is known to be hard, is there at least a good strategy that would give a small (if not the smallest) $n$ for many groups?

Note: I do not care whether $G$ acts transitively on $X$, so for example for $G=C_6$ the answer is $n=5$ (mapping the generator to (123)(45)), not $n=6$ (regular action).