  In this article, we give a sufficient and necessary condition for the
$C_2$-cofiniteness of the 2-cycle permutation orbifold model $(V\otimes
V)^\sigma$ for a $C_2$-cofinite vertex operator algebra and the 2-cycle
permutation $\sigma$ of $V\otimes V$. As an application, we show that the
2-cycle permutation orbifold model of the simple Virasoro vertex operator
algebra $L(c,0)$ of minimal central charge $c$ is $C_2$-cofinite.
