We investigate the spin-transfer torque in a magnetic multilayer structure by means of a spin-diffusion model. The torque in the considered system, consisting of two magnetic layers separated by a conducting layer, is caused by a perpendicular-to-plane current. We compute the strength of the fieldlike and the dampinglike torque for different material parameters and geometries. Our studies suggest that the fieldlike torque highly depends on the exchange-coupling strength of the itinerant electrons with the magnetization both in the pinned and the free layer. While a low coupling leads to very high fieldlike torques, a high coupling leads to low or even negative fieldlike torques. Furthermore, we demonstrate the significant impact of the fieldlike torque on the critical switching current of a magnetic multilayer. Thus, the dependence of the fieldlike torque on material parameters is considered very important for the development of applications such as spin-transfer-torque magnetic random-access memories and spin-torque oscillators.