TITLE: Smallest n for which G embeds in $S_n$?
QUESTION [33 upvotes]: 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).
Edit: If this is not specific enough, is there a method that could find the smallest $n$ (or one close to the smallest one) for any group of size $\le 10^7$ in 5 seconds on some computer algebra system?

REPLY [5 votes]: Let $\mu(G) = \min\{n \mid G \text{ embeds in } S_n\}$. Here are some results on $\mu(G)$ from this paper by O. Becker:

$\mu(G)$ is known for abelian groups.
It is known eactly when $\mu(G) = |G|$. If $\mu(G) < |G|$, then $\mu(G) \le \frac{5}{6}|G|$.
The identity $\mu(G\times H) = \mu(G) + \mu(H)$ holds for a wide family of groups, for instance - for all $G,H$ with central socle.