Representing the sporadic groups as noncentral automorphisms of \(p\)-groups (Q1814985)

By a result of Liebeck and the reviewer there is, for every finite group \(G\), a \(p\)-group \(L\) of class 2 such that \(G\) is isomorphic to \(\text{Aut}(L)/\text{Aut}_c(L)\) where \(\text{Aut}_c(L)\) is the group of central automorphisms. In the original construction \(L/L'\) becomes a free \(G\)-module, freely generated by \(k\) elements where \(k\) depends on the order and the number of the generators of \(G\). The authors show, that for the sporadic groups a modification is possible such that \(k=1\). This requires a special choice of generators for the groups \(G\) considered.