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We report on the experimental results of magnetic susceptibility, specificheat, electron spin resonance (ESR), and high-field magnetization measurementson a polycrystalline sample of the spin-$1/2$ distorted honeycomb-latticeantiferromagnet Cu$_2$(pymca)$_3$(ClO$_4$). Magnetic susceptibility shows abroad peak at about 25~K, which is typical of a low dimensionalantiferromagnet, and no long range magnetic order is observed down to 0.6~K inthe specific heat measurements. Magnetization curve up to 70~T at 1.4~K showstriple stepwise jumps. Assuming three different exchange bonds $J_{\rm A}$,$J_{\rm B}$ and $J_{\rm C}$ from the structure, the calculated magnetizationcurve reproduces the observed one when $J_{\rm A}/k_{\rm B} = 43.7~{\rm K}$,$J_{\rm B}/J_{\rm A} = 1$ and $J_{\rm C}/J_{\rm A} = 0.2$ except themagnetization near 70~T, where the observed magnetization indicates anotherstep while the calculated magnetization becomes saturated. The relationshipbetween magnetization plateaus and exchange bonds is discussed based on thenumerical calculations.

High-field Magnetism of the Honeycomb-lattice Antiferromagnet Cu$_2$(pymca)$_3$(ClO$_4$)