Computing quotients of group by elements of its lower exponent$-p$ central series

Question

Let $G$ be a finite $p−$group of number of generators $d$ and exponent$−p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G)=1$ in the series $$ G=P_0(G)≥...≥P_{i−1}(G)≥P_i(G)≥... $$ Where $P_i(G)=[P_{i−1}(G),G]P_{i−1}(G)^p$.

1/ Can you show me how to calculate $G/P_i(G)$´s using GAP system?

2/ Can you show me how to compute $G/P_1(G)$ using abelianisation and row-echelonisation (by hand)?

Thanks in advance

Answer 1

The series is in general called the $p$-central series, so the GAP command is PCentralSeries.

As for calculating it by hand, you might want to look at section 9.4.2 of Holt/Eick/O'Brien: Handbook of Computational Group Theory.