Same order statistics as abelian p-group not implies Lazard Lie group

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Statement

It is possible to have two finite p-groups G and H such that G is abelian, G and H are order statistics-equivalent (i.e., G and H have the same order statistics), but H is not a Lazard Lie group (?).

Related facts

Proof

There exist p-groups of arbitrarily large nilpotency class and exponent p.