The motion of N point vortices in a two-dimensional fluid is a Hamiltonian dynamical system with a 2N-dimensional phase space. The equations of motion for point vortices in a two-dimensional square doubly periodic domain are derived from those for an open domain. The Hamiltonian has three known constants of the motion and is thus believed to be nonintegrable for four or more vortices. Trajectories are numerically integrated from several initial conditions containing six vortices with varying total energy. Ergodicity on the surface defined by the constants of the motion is directly tested by comparing time-average and ensemble-average vortex pair statistics. It is found that the dynamics is not ergodic. There is evidence that the nonergodicity is not due to a gross fragmentation of phase space as might result from a broken symmetry. Vortex pair statistics are also used to test the randomness of the chaotic motion. It is found that the time-averaged statistics of the vortices are clearly distinct from those of independent random walkers.