The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid (AMG) for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)×(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are the monolithic classical AMG method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical AMG method is applied to solve the subsystems which originate in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We take advantage of representative one- and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, outperform the frequently-used AMG preconditioner and scale well both algorithmically and in parallel.