Large \(p\)-groups without proper subgroups with the same derived length. (Q2806142)

In the Sylow subgroup \(G_d=\Sigma_{p^d}\) of the symmetric group \(S_{p^d}\), \(d>1\), a subgroup \(H_d\) is constructed such that (1) the derived length \(\text{dl}(H_d)\) of \(H_d\) is equal to \(d\), (2) the minimal number of generators \(\text{d}(H_d)=d\), (3) if \(A<H_d\), then \(\text{dl}(A)<d\). It appears that \(H_d\) is a transitive subgroup of the symmetric group \(S_{p^d}\) of degree \(p^d\). Next, \(|\text{Z}(H_d)|=p\). Obviously, if \(d=2\), then \(H_d\) is minimal nonabelian. The class of \(H_d\) equals \(2^{d-1}\). The derived length of any proper epimorphic image of \(H_d\) is \(<d\). It appears that if \(p=2\), then \(H_d=G_d\). If \(p>2\), then \(|H_d|\) is much smaller then \(|G_d|\). If \(p=3\), then \(|H_d|=\frac12[\frac12(3^d-1)+d]\). A number of other results on the structure of \(H_d\) is proved.