Hyperspectral remote sensing is developing rapidly and widely used in various areas. However, hyperspectral images are always disturbed by noise, and the hot topic about how to denoise the hyperspectral data cube arises. This paper takes the challenge and proposes a denoising method based on group sparse representation by analyzing the characteristics of hyperspectral data. The proposal takes into account the correlation of different spectral bands and denoise the hyperspectral data effectively by grouping, training adaptive dictionary and sparse coding. In this paper, a fast clustering method is researched and a group sparse coding method based on improved Orthogonal Matching Pursuit is proposed. Experiments prove that the proposed algorithm suppresses the noise in the hyperspectral image and achieves high peak signal to noise ratio with high speed.