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

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 fact about::Lazard Lie group.

Related facts

 * Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring
 * Finite abelian groups with the same order statistics are isomorphic
 * Order statistics-equivalent not implies 1-isomorphic
 * Finite group having the same order statistics as a cyclic group is cyclic

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