Summary: We consider the quotient variety associated to a linear representation of the cyclic group of order \(p\) in characteristic \(p>0\). We estimate the minimal discrepancy of exceptional divisors over the singular locus. In particular, we give criteria for the quotient variety being terminal, canonical and log canonical. As an application, we obtain new examples of non-Cohen-Macaulay terminal singularities, adding to examples recently announced by Totaro.