The effect of the local symmetry on the magnetic properties of small Nickel clusters (up to 13 atoms) is investigated and compared with Pd and Rh equivalent structures. By means of self-consistent field/configuration interaction (SCF/CI) calculations of the intermediate neglect of differential overlap (INDO) type, the detailed geometry and multiplicity of the metal structures were determined. Low multiplicities in 5-fold symmetries are found to be more stable for the 13-atom clusters and can be described as the result of a spin quenching that follows a Jahn-Teller distortion of the higher symmetric icosahedral structures. The association of the magnetic properties of quasicrystals to the icosahedral symmetry is discussed.