A systematic investigation of the magnetic properties using AC susceptibility measurements has been performed in (Dy \(_{0.6}\) Gd \(_{0.4}\) ) \(_{5}\) Pd \(_{2}\) compound. The \(\chi '\) curve showed a peak \(T_{g1}\approx \) 59 K and a weak hump-like behavior at low temperatures. Interestingly, the \(\chi ''\) curve displayed two frequency-dependent distinct peaks, one at \(T_{g1}\) and other at \(T_{g2}\approx \) 16 K. An obtained value of the relative shift in freezing temperatures \(\delta T_{f1}\approx \) 0.017 and \(\delta T_{f2}\approx \) 0.07 are obtained from the AC susceptibility data reflects the formation of double cluster-glass states. The frequency dependence of T \(_f\) is also analyzed within the framework of dynamic scaling laws such as power law and Vogel-Fulcher law. The analysis using power law yields a characteristic time constant for a single spin flip is \(\tau ^*=\) \(1.55 \times 10^{-9}\) s and critical exponent \(z\nu '=\) 3.53 for the temperature \(T_{g1}\) . Whereas \(\tau ^*=\) \(3.98 \times 10^{-6}\) s and critical exponent \(z\nu '=\) 1.89 for the temperature \(T_{g2}\) . Further evidence of cluster-glass behavior comes from the frequency dependence of the freezing temperature fitted with the Vogel-Fulcher law. Values of fitting parameters are, Vogel-Fulcher temperature T \(_0 \approx 54.38\) K, an activation energy \( E_a / k_B\approx 69.18 \) K and \( E_a / k_BT_0\approx 1.27 > 1 \) for \(T_{g1}\) ; T \(_0 \approx 14.24\) K, \( E_a / k_B\approx 14.71 \) K, and \( E_a / k_BT_0\approx 1.03 > 1 \) for \(T_{g2}\) . The nonzero values of T \(_0\) and \( E_a / k_BT_0 \) further support the evidence for the cluster-glass behavior. The magnetic contribution to the specific heat follows a \( T^{3/2} \) temperature dependence below the cluster-glass freezing temperature, also supporting the evidence for cluster-glass behavior.
