Group classification of dynamics equations of self-gravitating gas

•The group classification problem for self-gravitating gas is solved.•The kernel of symmetry algebras is found.•All specifications of the parameter are retrieved.•The extensions of the kernel for all specifications are presented.•A comparison with the group classification of gas dynamics is done. In the paper, a group classification problem is solved for a system of equations which describes motion of self-gravitating gas. A parameter in group classification problem is a function which is determined by an equation of state. A kernel of Lie algebras admitted by the system and an algebra of equivalence transformations group are derived. All specifications of the parameter that lead to extensions of the kernel are found.