The Laplace operator \(\nabla^{2}\) in the \(5\)-dimesional space is represented as the sum of its radial part \(\beta^{-4}\frac{\partial}{\partial\beta}( \beta^{4}\frac{\partial}{\partial\beta}) \) and its angular part \(-\Lambda/\beta^{2}\). Let \(Y_{\lambda\alpha IM}\) denote the eigenfunctions of the operator \(\Lambda\) belonging to the eigenvalue \(\lambda( \lambda+3) \), where \(I,M\) are the angular-momentum labels and \(\alpha\) is an additional parameter. In [\textit{E. Chacón, M. Moshinsky} and \textit{R. T. Sharp}, J. Math. Phys. 17, 668--676 (1976) and 18, 870 (1977)], an explicit form for the eigenfunctions \(Y_{\lambda\alpha IM}\) is derived. The authors use a new basis to re-derive the results of [loc. cit.] more directly in terms of Legendre functions, by using a certain recurrence relation. The case of large \(\lambda\) and and the role of the octahedral symmetry group is discussed. The case of 5-dimensional space is important because some properties of atomic nuclei can be interpreted as rotations and vibrations of a quadrupole shape, which is described by five parameters.