Is my proof correct?
If $$1+1+2+2+2+2$$ was a class equation of some group $G$, then for some $x\in G$, $|Cl_G(x)|=2$ and as such, $|C_G(x)|=5$. But, from the class equation we have that $|Z(G)|=2$. As $Z(G)$ is a subgroup of $C_G(x)$, this would mean that $2$ divides $5$ a contradiction.
Is this correct?