We have performed magnetization measurements at high magnetic fields of up to 53 T on single crystals of a uranium heavy-fermion compound U$_2$Zn$_{17}$ grown by the Bridgman method. In the antiferromagnetic state below the N{e}el temperature $T_{rm N}$ = 9.7 K, a metamagnetic transition is found at $H_c$ $simeq$ 32 T for the field along the [11$bar{2}$0] direction ($a$-axis). The magnetic phase diagram for the field along the [11$bar{2}$0] direction is given. The magnetization curve shows a nonlinear increase at $H_m$ $simeq$ 35 T in the paramagnetic state above $T_{rm N}$ up to a characteristic temperature $T_{{chi}{rm max}}$ where the magnetic susceptibility or electrical resistivity shows a maximum value. This metamagnetic behavior of the magnetization at $H_m$ is discussed in comparison with the metamagnetic magnetism of the heavy-fermion superconductors UPt$_3$, URu$_2$Si$_2$, and UPd$_2$Al$_3$. We have also carried out high-pressure resistivity measurement on U$_2$Zn$_{17}$ using a diamond anvil cell up to 8.7 GPa. Noble gas argon was used as a pressure-transmitting medium to ensure a good hydrostatic environment. The N{e}el temperature $T_{rm N}$ is almost pressure-independent up to 4.7 GPa and starts to increase in the higher-pressure region. The pressure dependences of the coefficient of the $T^2$ term in the electrical resistivity $A$, the antiferromagnetic gap $Delta$, and the characteristic temperature $T_{{rho}{rm max}}$ are discussed. It is found that the effect of pressure on the electronic states in U$_2$Zn$_{17}$ is weak compared with those in the other heavy fermion compounds.
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