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 $\mathrm{\mid }Cent(G)\mathrm{\mid }=n$ and primitive $n$-centralizer if $\mathrm{\mid }Cent(G)\mathrm{\mid }=\mathrm{\mid }Cent(\frac{G}{Z(G)})\mathrm{\mid }=n$. In this paper, we characterize the $9$-centralizer and the primitive $9$-centralizer groups.