The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in one-dimensional slab geometry. The proposed method is defined by a multilevel system of equations that includes multigroup and effective one-group low-order NDA equations. The eigenvalue is evaluated in an exact projected solution space of the smallest dimensionality. Numerical results that illustrate the performance of the new algorithm are demonstrated.