Conjugacy and the Class Equation · Hard problem

In $S_9$, let $\sigma$ have cycle type $(4,3,2)$ and let $\rho$ have cycle type $(5,3,1)$ (a $5$-cycle, a $3$-cycle, and one fixed point). Is $|\operatorname{cl}(\sigma)|\cdot|C_{S_9}(\sigma)|$ the same number as $|\operatorname{cl}(\rho)|\cdot|C_{S_9}(\rho)|$? If so, what is that common value?