  A detailed study of the structural and magnetic properties of the spin-$3/2$
hyperkagome lattice compound Li$_2$MgMn$_3$O$_8$ is reported. This material
shows ferromagnetic response below $T_{\rm C} \simeq 20.6$ K, the temperature
almost three times lower than the Curie-Weiss temperature $\theta_{\rm CW}
\simeq 56.6$ K. Density-functional band-structure calculations suggest that
this reduction in $T_{\rm C}$ may be caused by long-range antiferromagnetic
couplings that frustrate nearest-neighbor ferromagnetic couplings on the
hyperkagome lattice. Large magnetocaloric effect is observed around the $T_{\rm
C}$ with a maximum value of isothermal entropy change $\Delta S_{\rm m}\simeq
20$ J/kg-K and a maximum relative cooling power of $RCP\simeq 840$ J/kg for the
7 T magnetic field change. Critical analysis of the magnetization data and
scaling analysis of the magnetocaloric effect suggest the 3D Heisenberg/XY
universality class of the transition. The DC and AC magnetization measurements
further reveal glassy nature of the ferromagnetic transition. A detailed study
of the non-equilibrium dynamics via magnetic relaxation and memory effect
measurements demonstrates that the system evolves through a large number of
intermediate metastable states and manifests significant memory effect in the
cluster-glass state.
