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 $C(G)le betasqrt{|G|}$ 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 $C(G)le (1+epsilon)sqrt{|G|}+c_{epsilon}$.