the parametrization of a Gumbel in terms of a Gaussian

Question

Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $F(x)^n$ (where $n$ is the number of observations) and inverse cumulative error functions, where $\alpha$ and $\beta$ are $-\sqrt{2} \text{ erfc}^{-1}\left(2-\frac{2}{n}\right)$ and $\sqrt{2} \left(\text{erfc}^{-1}\left(2-\frac{2}{n}\right)-\text{erfc}^{-1}\left(2-\frac{2}{e n}\right)\right)$ respectively. Thanks in advance.