Consider $D \in SU(n)$ ($n$ a multiple of 4), a diagonal matrix with values $\pm 1$ on the diagonal and with trace 0 (only possible for $n$ a multiple of 4). There are $n \choose n/2$ such matrices.
What is the dimension of the conjugacy class $\left\{Ad_{V}(D) \ | \ V \in SU(n) \right\}$. of such a $D$?