It is observed that every finite group is isometric to the group of Automorphisms of some Compact Riemann surface of genus 􀝃(≥ 2). Also found that the group of symmetries of a non-linear molecule (point group) is an excellent example of finite groups. In this paper we have considered the point group of Sulphur molecule S8 , which is a group of order 16. Then we prove that the bounds of the order of symmetry group of Automorphisms of Compact Riemann Surface on which the point group of S8 acts as a group of Automorphism is 4(g1), and the corresponding minimum value of g is 5, associated Fuchsian Group of signature is (2,2,2,2,2).