  A characterization of the quotients of $p$-class tower groups of quadratic
fields by terms in the lower $p$-central series plays an important role in the
formulation of conjectures by Boston, Hajir and the author about the
distribution of such groups as the base field varies. In this paper, another
equivalent criterion is given which resolves an issue that arose as part of the
group-theoretical calculations carried out in relation to these conjectures.
