For a finite non cyclic group G, let γ(G) be the smallest integer
k such that G contains k proper subgroups H1​,…,Hk​ with the
property that every element of G is contained in Hig​ for some i∈{1,…,k} and g∈G. We prove that if G is a noncyclic permutation
group of degree n, then γ(G)≤(n+2)/2. We then investigate the
structure of the groups G with γ(G)=σ(G) (where σ(G) is
the size of a minimal cover of G) and of those with $\gamma(G)=2.