Distribution of higher powers than 2 of a gaussian distribution

If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the Gamma Distribution parameters relate to the square part of $\chi^2$.

In particular I'm trying to calculate $X^4$, where $X \sim \mathcal{N} \left(0,\frac{1}{N} \right)$. How do you even take on such a problem?

Thanks for any tips!

For every positive $a$, the distribution of $T=|X|^a$ has density $$ \frac2{a\sqrt{2\pi}}t^{(1/a)-1}\exp\left(-\tfrac12t^{2/a}\right)\,\mathbf 1_{t>0}. $$ This follows from the usual change of variables method explained there.

For example, the distribution of $Z=X^4$ has density $$ \frac{1}{2\sqrt{2\pi}z^{3/4}}\exp\left(-\tfrac12\sqrt{z}\right)\,\mathbf 1_{z>0}. $$