In 2007, Cardona and Quer studied curves of genus 2 whose automorphism groups are isomorphic to two non-maximal groups   and  of orders 16 and 24, see [1]. They did not, however,  refer to maximal automorphism groups of genus 2. In this paper, we investigate distinct types of maximal groups of genus two in complete detail. We prove that there are exactly four types of maximal groups  of genus two and give their presentations as finitely presented, transitive permutation representations of certain degrees. Furthermore, their character tables, matrix representations, and their primary invariants and their Cayley color graphs and/or their Shreier’s coset graphs are found. We also compute the equations of the hyperelliptic curves covered by these four types of groups, which happen to be the groups obtained from the maximal automorphism bound for the soluble automorphism groups  supersoluble automorphism groups  nilpotent automorphism groups  and the quaternion group -group of order 8 for genus

Riemann surfaces, automorphism groups, hyperelliptic curves of genus 2, maximal soluble groups, maximal soluble or nilpotent automorphism groups, genus of a curve, the action of groups, maximal order of automorphism groups, presentations of groups in generators and relations, permutation representation of groups, Cayley graphs.