1-isomorphic to abelian p-group not implies Lazard Lie group

Statement
It is possible to have a group of prime power order that is 1-isomorphic to an abelian group of prime power order but is not a Lazard Lie group.

Related facts

 * Order statistics-equivalent not implies 1-isomorphic
 * Logarithm map from Lazard Lie group to its Lazard Lie ring is a 1-isomorphism
 * Same order statistics as abelian p-group not implies Lazard Lie group
 * Finite abelian groups with the same order statistics are isomorphic
 * Order statistics of a finite group determine whether it is nilpotent
 * Finite group having the same order statistics as a cyclic group is cyclic