The lattice model of the Coulomb glass in two dimensions with box-type random field distribution is studied at zero temperature for system size up to 962. To obtain the minimum energy state we annealed the system using Monte Carlo simulation followed by further minimization using cluster flipping. The values of the critical exponents are determined using the standard finite size scaling. We found that the correlation length ξ diverges with an exponent ν=1.0 at the critical disorder Wc=0.2253 and that χdis≈ξ4-η with η=2 for the disconnected susceptibility. The staggered magnetization behaves discontinuously around the transition and the critical exponent of magnetization β=0. The probability distribution of the staggered magnetization shows a three peak structure which is a characteristic feature for the phase coexistence at first-order phase transition. In addition to this, at the critical disorder we have also studied the properties of the domain for different system sizes. In contradiction with the Imry-Ma arguments, we found pinned and noncompact domains where most of the random field energy was contained in the domain wall. Our results are also inconsistent with Binder's roughening picture.