Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S\_n$.
Equivalently this is the number of ways that $G$ can act on the set $\{1,2,\ldots,n\}$.
Let $a\_{G,n}$ be the number of homomorphisms from $G$ to $S\_n$.
Let $\alpha(G) = \lim\_n \frac{\log(a\_{G,n})}{n\log n}$.
How can we determine $\alpha(G)$?


For example, if $G = \mathbb Z$ we have $a\_{G,n} = n!$, since a homomorphism $\mathbb Z \to S\_n$ is determined by the image of its generator. By Stirling's approximation we have $\alpha(\mathbb Z) = 1$.
More generally, for $G = F\_s$ the free group on $s$ generators, we get $\alpha(F\_s) = s$.
For $G = \mathbb Z/2$, a morphism $G\to S\_n$ can be seen as an order 2 element in $S\_n$.
We can create such a permutation by dividing the elements of $\{1,2,\ldots,n\}$ in to pairs, which can be done in $\frac{n!}{(n/2)!2^{n/2}}$ ways.
This is asymptotically $\exp(\frac12 n\log n)$.
There are some more permutations of order 2 that have fixed points, but not enough to change the asymptotic result.
So $\alpha(\mathbb Z/2) = \frac12$.
More generally, for any finite group $G$ we have $\alpha(G) = 1-\frac1{|G|}$.


How can we determine $\alpha(G)$ for an arbitrary group?



Asymptotic number of permutation representations of a given group
In general it is not known whether $\alpha(G)$ exists as a limit (as opposed to limsup). For virtually solvable groups of finite rank (whose subgroup growth is at most polynomial) the relation between subgroup growth and representation growth given in the comments shows that $\alpha(G) \le 1$ (and equals it if the group is residually finite).


For finite groups $\alpha(G) = 1 - \tfrac 1{|G|}$ (this is likely provable by elementary means, much more precise results on the asymptotic behaviour of $a\_{G, n}$ in this case are given by Müller <https://zbmath.org/0862.20019>).


An opposite situation is that of so-called large groups (which have a finite-index subgroup surjecting onto the free group $F\_2$). For these $\alpha(G) > 0$ if it exists, and it has been computed exactly for a few families, for example:


* Fuchsian groups (e.g. surface groups) where $\alpha(G) = -\chi(G)$ (see <https://zbmath.org/1059.20021>, <https://zbmath.org/1068.20052>)
* Right-angled Artin groups where $\alpha(G)$ equals the independence number of the defining graph (see <https://zbmath.org/1481.20094>)


There also are relevant sporadic examples in another paper of Müller--Schlage-Puchta (<https://zbmath.org/1127.20022>). In particular they observe that the 2-generator 1-relator group
$$ G = \langle a, b | a^mb^n \rangle $$
admits the free product $C\_m \* C\_n$ as a quotient and hence it satisfies that $\alpha(G) \ge 2 - \tfrac 1n - \tfrac 1m$ (more generally you can get 1-relator $d$-generated groups with $\alpha(G)$ arbitrarily close to $d$).


