Strong spin-lattice coupling and prominent frustration effects observed in the 50$%$ Fe-doped frustrated hexagonal ($h$)LuMnO$_3$ are reported. A N{e}el transition at $T_{mathrm N} approx$ 112~K and a possible spin re-orientation transition at $T_{mathrm {SR}} approx$ 55~K are observed in the magnetization data. From neutron powder diffraction data, the nuclear structure at and below 300~K was refined in polar $P6_3cm$ space group. While the magnetic structure of LuMnO$_3$ belongs to the $Gamma_4$ ($P6_3cm$) representation, that of LuFe$_{0.5}$Mn$_{0.5}$O$_3$ belongs to $Gamma_1$ ($P6_3cm$) which is supported by the strong intensity for the $mathbf{(100)}$ reflection and also judging by the presence of spin-lattice coupling. The refined atomic positions for Lu and Mn/Fe indicate significant atomic displacements at $T_{mathrm N}$ and $T_{mathrm {SR}}$ which confirms strong spin-lattice coupling. Our results complement the discovery of room temperature multiferroicity in thin films of $h$LuFeO$_3$ and would give impetus to study LuFe$_{1-x}$Mn$_x$O$_3$ systems as potential multiferroics where electric polarization is linked to giant atomic displacements.
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