We report the crystal growth and structural and magnetic properties of quasi two-dimensional $S=1/2$ quantum magnet Cu[C$_6$H$_2$(COO)$_4$][H$_3$N-(CH$_2$)$_2$-NH$_3$]$cdot$3H$_2$O. It is found to crystallize in a monoclinic structure with space group $C2/m$. The CuO$_4$ plaquettes are connected into a two-dimensional framework in the $ab$-plane through the anions of [C$_6$H$_2$(COO)$_4$]$^{4-}$ (pyromellitic acid). The [H$_3$N-(CH$_2$)$_2$-NH$_3$]$^{2+}$$cdot$3H$_2$O groups are located between the layers and provide a weak interlayer connection via hydrogen (H...O) bonds. The temperature dependent magnetic susceptibility is well described by $S=1/2$ frustrated square lattice ($J_1-J_2$) model with nearest-neighbor interaction $J_1/k_{rm B} simeq 5.35$ K and next-nearest-neighbor interaction $J_2/k_{rm B} simeq -0.01$ K. Even, our analysis using frustrated rectangular lattice ($J_{1a,b}-J_2$) model confirms almost isotropic nearest-neighbour interactions ($J_{rm 1a}/k_{rm B} simeq 5.31$ K and $J_{rm 1b}/k_{rm B} simeq 5.38$ K) in the $ab$-plane and $J_2/k_{rm B}simeq-0.24$ K. Further, the isothermal magnetization at $T=1.9$ K is also well described by a non-frustrated square lattice model with $J_1/k_{rm B} simeq 5.2$ K. Based on the $J_2/J_1$ ratio, the compound can be placed in the N{e}el antiferromagnetic state of the $J_1 - J_2$ phase diagram. No signature of magnetic long-range-order was detected down to 2 K.
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