Closed form of product of Gamma function

Workaround:

$$\Gamma \left(\frac{k}{n}\right)=\frac{\Gamma \left(\frac{k}{n}+1\right)}{\frac{k}{n}}$$

$Version
(* "12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)" *)

Product[Gamma[k/n + 1]/(k/n), {k, 1, n - 1}]
(* (2 π)^(1/2 (-1 + n))/Sqrt[n] *)

The indeterminate can be overcome using the full identity for $\Gamma(nz)$:

$$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-1/2}\prod_{k=0}^{n-1}\Gamma(z+\frac{k}{n})$$

and taking the limit as $z\rightarrow 0$:

$Version
(* "12.0.0 for Linux x86 (64-bit) (April 15, 2019)" *)

Limit[Product[Gamma[z + k/n], {k, 1, n - 1}], z -> 0]
(* (2 \[Pi])^(1/2 (-1 + n))/Sqrt[n] *)