We report the structural, static, and dynamic properties of Cr$_{0.5}$Fe$_{0.5}$Ga by means of powder x-ray diffraction, magnetization, heat capacity, magnetic relaxation, and magnetic memory effect measurements. DC magnetization and AC susceptibility studies reveal a spin-glass transition at around $T_{rm f} simeq 22$~K. An intermediate value of the relative shift in freezing temperature $delta T_{rm f} simeq 0.017$, obtained from the AC susceptibility data reflects the formation of cluster spin-glass states. The frequency dependence of $T_{rm f}$ is also analyzed within the framework of dynamic scaling laws. The analysis using power law yields a time constant for a single spin flip $tau* simeq 1.1times10^{-10}$~s and critical exponent $z u^{prime}=4.2pm0.2$. On the other hand, the Vogel-Fulcher (VF) law yields the time constant for a single spin flip $tau_0 simeq 6.6times10^{-9}$~s, VF temperature $T_{rm 0}=21.1pm0.1$~K, and an activation energy $E_{rm a}/k_{rm B} simeq 16$~K. The value of $tau*$ and $tau_0$ along with a non-zero value of $T_{rm 0}$ provide further evidence for the cluster spin-glass behaviour. The magnetic field dependent $T_{rm f}$ follows the de Almeida-Thouless line with a non-mean-field type instability, reflecting either a different universality class or strong anisotropy in the spin system. A detailed non-equilibrium dynamics study via relaxation and memory effect experiments demonstrates striking memory effects. All the above observations render a cluster spin-glass behaviour which is triggered by magnetic frustration due to competing antiferromagnetic and ferromagnetic interactions and magnetic site disorder. Moreover, the asymmetric response of magnetic relaxation with respect to the change in temperature, below $T_{rm f}$ can be explained by the hierarchical model.
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