  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 $\Gamma_4$ ($P6'_3c'm$)
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.
