Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division $\mathbb{Q}$-algebra $D$ is finite dimensional. What is its center?

I've heard that the center is the number field generated by the values of the character of an irreducible constituent of $\rho_{\mathbb{C}}$ but I cannot see this (and I don't know where to look this up).