Can this be the class-equation of a finite group $G$ of order 10? Can this be the class-equation of a finite group $G$ of order 10?

$10=1+1+2+2+2+2$

I know that if conjugacy class of an element has order one then it must belong to $Z(G)$ and vice versa. Here $o(Z(G))=2$ .Also order of the conjugacy class must divide order of the group .
Keeping these two facts in mind I feel the it must be true.however the answer is no .Why?

$G/Z(G)$ is cyclic group of order $5$ and hence it implies that $G$  is abelian see, which means $G=Z(G)$ but class equation tells us $|Z(G)|=2$. Hence not possible.
