Maximal subgroups of finite groups avoiding the elements of a generating set

We give an elementary proof of the following remark: if G is a finite group and {g1,…,gd} is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that M∩{g1,…,gd}=∅. This result leads us to investigate the freedom that one has in the choice of the maximal subgroup M of G. We obtain information in this direction in the case when G is soluble, describing for example the structure of G when there is a unique choice for M. When G is a primitive permutation group one can ask whether is it possible to choose in the role of M a point-stabilizer. We give a positive answer when G is a 3-generated primitive permutation group but we leave open the following question: does there exist a (soluble) primitive permutation group G=⟨g1,…,gd⟩ with d(G)=d>3 and with ⋂1≤i≤dsupp(gi)=∅? We obtain a weaker result in this direction: if G=⟨g1,…,gd⟩ with d(G)=d, then supp(gi)∩supp(gj)≠∅ for all i,j∈{1,…,d}