We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.