Let $G$ be a graph with adjacency matrix $A$. The transition matrix corresponding to $G$ is defined by $H(t):=\exp{\left(itA\right)}$, $t\in\Rl$. The graph $G$ is said to have perfect state transfer (PST) from a vertex $u$ to another vertex $v$, if there exist $��\in\Rl$ such that the $uv$-th entry of $H(��)$ has unit modulus. The graph $G$ is said to be periodic at $��\in\Rl$ if there exist $��\in\Cl$ with $|��|=1$ such that $H(��)=��I$, where $I$ is the identity matrix. A $\mathit{gcd}$-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. In this paper, we construct classes of $\mathit{gcd}$-graphs having periodicity and perfect state transfer.