Semi-extraspecial groups with an abelian subgroup of maximal possible order

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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.