On finite groups with nine centralizers

Given a finite group $G$, let $Cent(G)$ denote the set of distinct centralizers of elements of $G$. The group $G$ is called $n$-centralizer if $|Cent(G)|=n$ and primitive $n$-centralizer if $|Cent(G)|=|Cent(\frac{G}{Z(G)})|=n$. In this paper, we characterize the $9$-centralizer and the primitive $9$-centralizer groups.