Normal Distribution - variance of minimal MSE (mean squared error)

Question

We are given $n$ normally distributed random variables $X_1,\ldots, X_n$ with mean $\mu$ and variance $\sigma^2$. We have \begin{align} S^2 = C \sum_{i=1}^n (x_i - \bar{x})^2 \end{align} We want to find $C$, such that the variance has minimal MSE. Now, I already have that, \begin{align} E(S^2)=C \, (n-1) \,\sigma ^2, \end{align} but I don't see how I would calculate $\operatorname{var}(S^2)$ to find the $\operatorname{MSE}(S^2)$ (and minimize it with respect to $C$). Thank you in advance.