An upper bound on the Chebotarev invariant of a finite group

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ 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 $C(G)\le \beta\sqrt{|G|}$ for some absolute constant $\beta$. In this paper we show that, when $G$ is soluble, then $\beta$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $\epsilon>0$ there exists a constant $c_{\epsilon}$ such that $C(G)\le (1+\epsilon)\sqrt{|G|}+c_{\epsilon}$.