We study a random walk on the special unitary group SU(N) consisting of a product of matrices chosen Haar uniformly from a fixed conjugacy class. In particular, we make use of the representation theory of matrix Lie groups to show two results about the rate of convergence of the random walk's distribution to the Haar measure in total variation distance. We derive a lower bound in total variation distance before a threshold number of steps, which appears to be an example of a cut-off phenomenon, and for dimension N=2 we prove exponentially fast convergence.