Abstract: Incomplete invariant groups, group invariant solutions, explicit exact solutions and dynamic behavior characteristics of solutions of a class of integro-partial differential population balance equation were investigated in this work. Firstly, admitted scaling transformation Lie groups of the population balance equation were explored by using the methods of scaling transformation group analysis. Secondly, an integro-partial differential population ba-lance equation was transformed into a pure partial differential equation, complete admitted Lie groups of the pure partial differential equation were studied by using the method of classical Lie group analysis. Thirdly, incomplete admitted Lie groups of the original integro-partial differential population balance equation were verified by using the methods of developed Lie group analysis. Finally, incomplete invariant groups of the original integro-partial diffe-rential population balance equation were found. All group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions were also given. The related analysis for dynamic behavior characteristics of solutions with evolution of the size distribution were also presented.