A Klein surface with boundary of algebraic genus $\mathfrak{p}\geq
2$, has at most $12(\mathfrak{p}-1)$ automorphisms. The groups
attaining this upper bound are called $M^{\ast}$-groups, and the
corresponding surfaces are said to have maximal symmetry. The
$M^{\ast}$-groups are characterized by a partial presentation by
generators and relators.
The alternating groups $A_{n}$ were proved to be $M^{\ast}$-groups when
$n\geq 168$ by M. Conder. In this work we prove that $A_{n}$ is
an $M^{\ast }$-group if and only if $n\geq 13$ or $n=5,10$. In
addition, we describe topologically the surfaces with maximal
symmetry having $A_{n}$ as automorphism group, in terms of the
partial presentation of the group. As an application we determine
explicitly all such surfaces for $n\leq 14$.
Each finite group $G$ acts as an automorphism group of several
Klein surfaces. The minimal genus of these surfaces is called the
real genus of the group, $\rho(G)$. If $G$ is an $M^{\ast}$-group then
$\rho(G)=\frac{o(G)}{12}+1$. We end our work by calculating the real genus
of the alternating groups which are not $M^{\ast}$-groups.