Let p be a prime. A p-group G is defined to be semi-extraspecial if for
every maximal subgroup N in Z(G) the quotient G/N is a an extraspecial
group. In addition, we say that G is ultraspecial if G is semi-extraspecial
and ∣G:G′∣=∣G′∣2. In this paper, we prove that every p-group of
nilpotence class 2 is isomorphic to a subgroup of some ultraspecial group.
Given a prime p and a positive integer n, we provide a framework to
construct of all the ultraspecial groups order p3n that contain an abelian
subgroup of order p2n. In the literature, it has been proved that every
ultraspecial group G order p3n with at least two abelian subgroups of
order p2n can be associated to a semifield. We provide a generalization of
semifield, and then we show that every semi-extraspecial group G that is the
product of two abelian subgroups can be associated with this generalization of
semifield