  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 $\tau* \simeq 1.1\times10^{-10}$~s and critical exponent
$z\nu^{\prime}=4.2\pm0.2$. On the other hand, the Vogel-Fulcher (VF) law yields
the time constant for a single spin flip $\tau_0 \simeq 6.6\times10^{-9}$~s, VF
temperature $T_{\rm 0}=21.1\pm0.1$~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.
