Minimum Size of a Poset Realizing $\Z_{2}\times\Z_{2^{n}}$ as its Automorphism Group

We study the realization of finite groups as automorphism groups of finite posets.

Given a finite group $G$, let $\beta(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$.

While $\beta(G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open.

In this paper we prove that $\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$ for every $n\ge 3$.