  One of the most interesting problems on maximum nullity (minimum rank) is to
characterize $M(\mathcal{G})$ ($mr(\mathcal{G})$) for a graph $\mathcal{G}$. In
this regard, many researchers have been trying to find an upper or lower bound
for the maximum nullity. For more results on this topic, see \cite{4},
\cite{2}, \cite{10} and \cite{1}. In this paper, by using a result of Babai
\cite{Babai}, which presents the spectrum of a Cayley graph in terms of
irreducible characters of the underlying group, and using representation and
character of groups, we give a lower bound for the maximum nullity of Cayley
graph, $X_S(G)$, where $G=\langle a\rangle$ is a cyclic group, or $G=G_1\times
\cdots\times G_t$ such that $G_1=\langle a\rangle$ is a cyclic group and $G_i$
is an arbitrary finite group, for some $2\leq i\leq t$, with determine the
spectrum of Cayley graphs.
