We report on the experimental results of magnetic susceptibility, specific heat, electron spin resonance (ESR), and high-field magnetization measurements on a polycrystalline sample of the spin-$1/2$ distorted honeycomb-lattice antiferromagnet Cu$_2$(pymca)$_3$(ClO$_4$). Magnetic susceptibility shows a broad peak at about 25~K, which is typical of a low dimensional antiferromagnet, and no long range magnetic order is observed down to 0.6~K in the specific heat measurements. Magnetization curve up to 70~T at 1.4~K shows triple stepwise jumps. Assuming three different exchange bonds $J_{rm A}$, $J_{rm B}$ and $J_{rm C}$ from the structure, the calculated magnetization curve 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 the magnetization near 70~T, where the observed magnetization indicates another step while the calculated magnetization becomes saturated. The relationship between magnetization plateaus and exchange bonds is discussed based on the numerical calculations.
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