We have investigated structural and magnetic properties having cluster-glass (CG) like ordering at low temperature in a polycrystalline La_1.5Ca_0.5CoMnO₆ double perovskite sample. In this anti-site disordered double perovskite, ferromagnetic (FM) ordering is observed at ~157K. From temperature variation linear ac susceptibility measurements, a CG state is clearly observed below a freezing temperature of ~50.1K, the origin of which can be best explained through (1) structural disorder and (2) consequence of the coexistence and competition of antiferromagnetic and FM interactions. The linear complex ac susceptibility data near the freezing temperature are best fitted to a critical slowing down model characterized by the dynamic exponents zυ~4.95(5) and τ₀~1.23 10^-7 s. These values are very similar to what is generally observed in a CG system. The glassy transition temperature versus H^2/3_dcbehaviour follows the Almeida-Thouless line. The time dependent slow relaxation dynamics and aging effect have also been investigated to confirm a CG-like state formation in this compound.