Based on the Riemann fractional derivative the Casimir operators and multipletts for the fractional extension of the rotation group SO(n) are calculated algebraically. The spectrum of the corresponding fractional symmetric rigid rotor is discussed. It is shown, that the rotational, vibrational and $\gamma$-unstable limits of the standard geometric collective models are particular limits of this spectrum. A comparison with the ground state band spectra of nuclei shows an agreement with experimental data better than 2%. The derived results indicate, that the fractional symmetric rigid rotor is an appropriate tool for a description of low energy nuclear excitations.