Noncollinear antiferromagnets have promising potential to replace ferromagnets in the field of spintronics as high-density devices with ultrafast operation. To take full advantage of noncollinear antiferromagnets in spintronics applications, it is important to achieve efficient manipulation of noncollinear antiferromagnetic spin. Here, using the anomalous Hall effect as an electrical signal of the triangular magnetic configuration, spin-orbit torque switching with no external magnetic field is demonstrated in noncollinear antiferromagnetic antiperovskite manganese nitride Mn$_3$GaN at room temperature. The pulse-width dependence and subsequent relaxation of Hall signal behavior indicate that the spin-orbit torque plays a more important role than the thermal contribution due to pulse injection. In addition, multistate memristive switching with respect to pulse current density was observed. The findings advance the effective control of noncollinear antiferromagnetic spin, facilitating the use of such materials in antiferromagnetic spintronics and neuromorphic computing applications.