In order to further improve the fitting prediction accuracy of the grey prediction model,\ aiming at the limitation of the existing grey prediction model with time power term,\ a discrete grey model with fractional order $\gamma-$polynomial,\ abbreviated as FPDGM(1,1,$N$) model,\ is proposed by introducing fractional order polynomial.\ The modeling mechanism,\ parameter estimation and recursive time response formula of the model are studied.\ The properties of the model under some special values of $\gamma$ and $N$ are discussed.\ The results show that DGM(1,1) model, NDGM(1,1) model and DGM(1,1,$N$) model are all special forms of FPDGM(1,1,$N$) model.\ Therefore,\ this model formally unifies the existing grey model with time power term and enlarges the application scope of grey prediction theory.\ Finally,\ experiments show that the new model has better fitting and prediction accuracy,\ which verifies the validity and applicability of the proposed model.