  Let $\pi$ be a proper subset of the set of all primes. Denote by $r$ the
smallest prime which does not belong to $\pi$ and set $m = r$ if $r = 2$ or $3$
and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a
conjugacy class $D$ of a finite group $G$ is contained in the $\pi$-radical
$\mathrm{O}_\pi(G)$ of $G$ if and only if every $m$ elements of $D$ generate a
$\pi$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian
composition factors are isomorphic to sporadic or alternating groups.
