The formal framework for quantum mechanics is an infinite number of dimensional space. Hereby, in any analytical calculation of the quantum system, the energy eigenvalues and corresponding wave functions can be represented easily in a finite-dimensional basis set. In this work, the approximate analytical solutions of the hyper-radial Schrodinger equation are obtained for the generalized Wood-Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are derived for any angular momentum case by means of state-of-the-art Nikiforov-Uvarov and supersymmetric quantum mechanics methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transforming each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well V-0 and W, the radial n(r) and l orbital quantum numbers and parameters D, a, R-0 are also identified in detail. Next, the bound state energies and corresponding normalized hyper-radial wave functions for the neutron system of the Fe-56 nucleus are calculated in D = 2 and D = 3 as well as the energy spectrum expressions of other higher dimensions are revealed by using the energy spectrum of D = 2 and D = 3.