2 deleted 6 characters in body

I guess you are considering normal singularities. In this case your statement follows from you can probably use [Kollar-Mori, Birational geometry of algebraic varieties, Corollary 5.21 p. 161]. For the reader's covenience, let me write the complete statement.

Proposition. Let $x \in X$ be a germ of a normal singularity.

(1) $x \in X$ is Kawamata Log Terminal if and only if it is a cyclic quotient of an index $1$ canonical singularity $0 \in Y$ by an action which is fixed point free in codimension $1$ (that is, $\textrm{Sing}(X)$ has codimension at least $2$).

(2) If $x \in X$ is terminal (resp. canonical) then it is a cyclic quotient of an index $1$ terminal (resp. canonical) singularity $0 \in Y$ by an action which is fixed point free in codimension $2$ (that is, $\textrm{Sing}(X)$ has codimension at least $3$).

1

I guess you are considering normal singularities. In this case your statement follows from [Kollar-Mori, Birational geometry of algebraic varieties, Corollary 5.21 p. 161]. For the reader's covenience, let me write the complete statement.

Proposition. Let $x \in X$ be a germ of a normal singularity.

(1) $x \in X$ is Kawamata Log Terminal if and only if it is a cyclic quotient of an index $1$ canonical singularity $0 \in Y$ by an action which is fixed point free in codimension $1$ (that is, $\textrm{Sing}(X)$ has codimension at least $2$).

(2) If $x \in X$ is terminal (resp. canonical) then it is a cyclic quotient of an index $1$ terminal (resp. canonical) singularity $0 \in Y$ by an action which is fixed point free in codimension $2$ (that is, $\textrm{Sing}(X)$ has codimension at least $3$).