Specifying the sides is not enough. You probably mean that also the angles are all equal, so we are dealing with the regular pentagon of side $8$.
The centre of mass will be at the centre of the circle that the regular pentagon is inscribed in.
To describe the location of the centre, imagine that $AB$ is one of the edges, and let $O$ be the centre of the circle. Let $M$ be the midpoint of $AB$. To get to $O$, we draw the perpendicular bisector of $AB$, and go up a distance $d$ from $M$, where $d=4\tan(54^\circ)$.
Remark: One can get an explicit expression for the required $\tan(54^\circ)$ in terms of square roots if that is desired. It turns out that