The fact that the data sequence satisfies the grey exponential law is the modeling condition of the grey prediction model. The grey system theory weakens the randomness of data by accumulating generation method. The fractional order accumulating generation operator is the basis for establishing the fractional order grey prediction model. By the Gamma function, the detailed proof steps of the integer order accumulating generation operator and fractional order accumulating generation operator based on the first order accumulating generation operator are given. The properties of fractional grey accumulating generation operator that satisfy the important properties such as fixed point theorem, information priority principle, exchange law and exponential law are proved. The data changing trend of the fractional order accumulating generation operator is given by numerical example. The properties of fixed point theorem, commutative law and exponential law of fractional order grey accumulating generator are verified by numerical example. The research method and properties are helpful for the theory of fractional order grey operators and fractional order grey prediction model.