Probabilistic construction of some extremal $p$-groups

A $p$-group $G$ is called ab-maximal if $\mathrm{\mid }H:H^{\mathrm{\prime }}\mathrm{\mid }<\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }$ for every proper subgroup $H$ of $G$. Similarly, $G$ is called $d$-maximal if $d(H)<d(G)$ for every proper subgroup $H$ of $G$, where $d(H)$ is the minimal number of generators of $H$. If $G$ is ab-maximal then $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\ge p^3\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$, while if $G$ is $d$-maximal and $p\mathrm{\ne }2$ then $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\ge p^2\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$. Answering questions of González-Sánchez–Klopsch and Lisi–Sabatini, for all $p$ we construct infinitely many ab-maximal $p$-groups of class $2$ with $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }=p^3\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$, and infinitely many $d$-maximal $p$-groups of class $2$ with $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }=p^2\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$. The construction is probabilistic and based on the degeneracy of random alternating bilinear maps on subspaces. It is notable however that in the ab-maximal case we do not have a high-probability result but rather in a suitable sense the proportion of class-$2$ groups with $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }=p^n$ and $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }=p^{n-3}$ that are ab-maximal is close to $1\mathrm{/}e$.