Finite-time thermodynamics is used to determine the maximum power output and the corresponding thermal efficiency for an irreversible closed-cycle Brayton heat engine. A realistic model is presented, which includes three types of irreversibilities: finite thermal conductance between the working fluid and the reservoirs, heat leaks between the reservoirs, and internal irreversibility inside the Brayton heat engine. The effects of heat leaks, hot-to-cold reservoir temperature ratios, and component efficiencies on the maximum power output and the corresponding thermal efficiency are studied. With component efficiencies less than 100%, the optimum ratio of the hot-end heat exchanger conductance to the total conductance of the two heat exchangers is found to be less than 0.5. Moreover, this ratio increases as the component efficiencies and the total conductance of the two heat exchangers are increased, and decreases as the hot-to-cold reservoir temperature ratio is increased.