The power structure of finite p-groups satisfying conditions like those of Lazard

Lazard proved that finitely generated pro-p groups satisfying gamma(m)(G)less than or equal toG(p h) are p-adic analytic if m<p(h). Recent work has given a very thorough understanding of the finite p-groups that satisfy this condition with h=1. In this paper, we consider pro-p groups that show that those results cannot be extended to the case of h=2 for m >= 2p-1. We do extend many of the results to this case when m <= 2p-3 as well as under some weaker hypotheses.