Existence at n=6 of a class-2 exponent-p group H yielding ultraspecial groups with |A:Z(G)|=p^{n+2}

Determine whether, for n=6 and a prime p, there exists a class-2, exponent-p group H of order p^{2n} with |Z(H)|=p^{n+1} and centralizers C_H(h)=⟨h, Z(H)⟩ for all h ∈ H \ Z(H), which would yield (via the standard construction described) an ultraspecial p-group G of order p^{3n} possessing a maximal abelian subgroup of index |A:Z(G)|=p^{n+2}.

Background

Section 4 investigates semi-extraspecial (s.e.s.) and ultraspecial p-groups, focusing on the sizes of maximal abelian subgroups. The authors explain that if one can construct a class-2, exponent-p group H of order p^{2n} satisfying |Z(H)|=p^{n+1} and C_H(h)=⟨h, Z(H)⟩ for all h outside the center, then an ultraspecial group G of order p^{3n} with a maximal abelian subgroup of size p^{n+2} can be obtained.

They provide explicit constructions for n=4 and n=5 but note uncertainty at n=6, leaving open the existence of such an H (and therefore of the corresponding ultraspecial G) in that case.