How to compute the fundamental group of a specific identification space of the unit disk

Question

Say that $X_n$ is the space defined by taking the unit disk in the complex plane (that is $\{z||z|\le 1\}$) and identifying points on the boundary of the disk as follows: if $|z|=1$ then we identify $z$ with $z\cdot e^{\frac{2\pi i}{n}}$ (this means that each point on the boundary will be identified with $n-1$ other points and that points on the interior will not be affected). I am supposed to compute the fundamental group of this space but I am not sure how to proceed. I tried to do some easy cases. n=2 for example yields $\mathbb{Z}/2\mathbb{Z}$ since it's another theorem that the unit disk with antipodal points identified is homeomorphic to $\mathbb{R}P^2$ which has fundamental group $\mathbb{Z}/2\mathbb{Z}$. I think that in general it is $\mathbb{Z}/n\mathbb{Z}$. Do I need to use Seifert Van Kampmen's theorem for this? If not, is there some way to directly compute it?