Recent progress in solving quadratic assignment problems (QAPs) from the QAPLIB (Quadratic Assignment Problem Library) test set has come from mixed-integer linear or quadratic programming models that are solved in a branch-and-bound framework. Semidefinite programming (SDP) bounds for QAPs have also been studied in some detail, but their computational impact has been limited so far, mostly because of the restrictive size of the early relaxations. Some recent progress has been made by studying smaller SDP relaxations and by exploiting group symmetry in the QAP data. In this work, we introduce a new SDP relaxation, where the matrix variables are only of the order of the QAP dimension, and we show how one may exploit group symmetry in the problem data for this relaxation. We also provide a detailed numerical comparison with related bounds from the literature. In particular, we compute the best-known lower bounds for two QAPLIB instances.