TITLE: Smallest permutation representation of a finite group?
QUESTION [33 upvotes]: Given a finite group G, I'm interested to know the smallest size of a set X such that G acts faithfully on X. It's easy for abelian groups - decompose into cyclic groups of prime power order and add their sizes. And the non-abelian group of order pq (p, q primes, q = 1 mod p) embeds in the symmetric group of degree q as shown here: www.jstor.org/stable/2306479.
How much is known about this problem in general?

REPLY [4 votes]: Although tighter results may be available for specific classes of groups (as in the comments and other answer), Babai, Goodman, and Pyber [1] showed that in some sense - made precise in the result below - the only obstacle to having a small faithful permutation representation is having a prime-power cyclic group of small index.
Quoting the main result from the abstract:

Let $\mu(G)$ be the smallest degree of a faithful permutation representation of $G$, and let $i(G)$ be the index of its largest cyclic subgroup of prime-power order. Then $i(G) \geq |G| / \mu(G) \geq \exp(c \sqrt{\log i(G)})$ for some constant $c > 0$.

Rephrasing in terms of bounds on $\mu(G)$, this gives
$$ |G|/\exp(c \sqrt{\log i(G)}) \geq \mu(G) \geq |G| / i(G)$$
[1] László Babai, Albert J. Goodman & László Pyber. On faithful permutation representations of small degree. Comm. Alg., 21(5):1587-1602, 1993. doi:10.1080/00927879308824639