A model of compact nuclei of galaxies as spherically-symmetric star clusters is proposed. A concept of the equation of state for star clusters in statistical equilibrium is introduced (galactic nuclei are systems in statistical equilibrium if their age is of the order of the age of the Universe). It is shown that a statistically equilibrium star cluster is described by the equation of state of a polytrope P = aρ3, and with its help the main parameters of compact nuclei of galaxies are calculated. The formula M = 2.524GR5/a for mass M and radius R of the cluster is derived.