Abstract: The point groups of octagonal system in quasicrystal are studied. By using stereographic projection of each point group, all symmetry operation and generating operation of the point groups are listed, and group multiplication table of maximal proper point group 822 is filled in. In octagonal coordinate system defined by us, the matrixes of the point groups symmetry operation are derived. There are thirty-two 3 × 3 matrixes. The marixes form are compact, their matrix element possible values are ：0, ± 1, ±√2. Therein,√2 is an especial irrational number,which indicated the quasi-periodicity of octagonal system quasicrystal.

Key words: quasicrystal, octagonal system, point group, stereographic projection, group multiplication, matrix