A compact bordered Klein
surface of genus g ≥ 2 is said to have maximal symmetry if its automorphism group is
of order 12(g − 1), the largest possible. For each value of the positive integer g there
are, of course, several different topological types of bordered surfaces of genus g; each
distinct topological type is called a species of the genus g. Here we classify the species
of bordered Klein surfaces with maximal symmetry of genus g ≤ 40; there are 32
species in 18 different genera. We also classify the species with maximal
symmetry that have no more than 5 boundary components. To aid in the
classification two group-theoretic constructions that give new surfaces with
maximal symmetry and a family of M∗-groups are introduced. We also establish
several general results about the species of a surface with maximal symmetry.
In particular we show that if X is a non-orientable bordered surface with
maximal symmetry and solvable automorphism group, then the genus of X is
odd.
