Let $G$ be a finite group and let $Z(G)$ denote its center. A simple result states that if $G/Z(G)$ a nontrivial cyclic group then $G$ is abelian. Of course if $G$ is abelian then $Z(G)=G$ and consequently $G/Z(G)$ is the trivial group. Hence we conclude that no non-trivial cyclic groups can result when we quotient a group by its center. My question is whether there are any other restrictions as to what groups can occur as the quotient of some group by its center.
A group that can be written as $G/Z(G)$ for some group $G$ is called 'capable'. It appears to be easy to find necessary conditions for this, but difficult to find sufficient ones. The problem of finding such groups has been deemed 'interesting'. The question of which finitely generated abelian groups are capable was answered by Baer in 1938. Check out this, from our own @Arturo Magidin.