  For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer
$k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the
property that every element of $G$ is contained in $H_i^g$ for some $i \in
\{1,\dots,k\}$ and $g \in G.$ We prove that if $G$ is a noncyclic permutation
group of degree $n,$ then $\gamma(G)\leq (n+2)/2.$ We then investigate the
structure of the groups $G$ with $\gamma(G)=\sigma(G)$ (where $\sigma(G)$ is
the size of a minimal cover of $G$) and of those with $\gamma(G)=2.$
