Let $G$ be a finite group and $Z(G)$ be the center of $G$. For a subset $A$ of $G$, we define $k_G(A)$, the number of conjugacy classes of $G$ which intersect $A$ non-trivially. In this paper, we verify the structure of all finite groups $G$ which satisfy the property $k_G(G-Z(G))=5$ and classify them.
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