While there has been a surge of articles on convergence diagnostic tools for MCMC on continuous stationary distributions and ordinal state spaces, Bayesian clustering has spawned demands for tools designed specifically for nominal finite state spaces - grouping space. To fill this gap we propose a simple quantitative convergence criterion for MCMC algorithms run on nominal state spaces that has an intuitive interpretation which is a one-dimensional goodness-of-fit statistic. We study the asymptotic behaviour of the statistic and estimate its variance using the regenerative simulation. The convergence assessment is performed via a formal statistical significance test. We study the performance of the proposed criterion via simulation. We finally consider the particular application of clustering of genetic mutants of the flowering plant Arabidopsis thaliana .