An upper bound on the Chebotarev invariant of a finite group

A subset $\{g_1,\dots ,g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1},\dots ,g_d^{x_d}\}$ generates $G$ for every choice of $x_i\in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)\le \beta \sqrt{\mathrm{\mid }G\mathrm{\mid }}$ for some absolute constant $\beta$. In this paper we show that, when $G$ is soluble, then $\beta$ is at most $5\mathrm{/}3$. We also show that this is best possible. Furthermore, we show that, in general, for each $ϵ>0$ there exists a constant $c_{ϵ}$ such that $C(G)\le (1+ϵ)\sqrt{\mathrm{\mid }G\mathrm{\mid }}+c_{ϵ}$.