  We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are
extensions of the $N=1$ superconformal algebra by a primary supercurrent of
conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the
Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field
realization of the algebra in terms of the screening operators. At central
charge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to the
superconformal algebra associated to sigma models based on eight-dimensional
manifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa $Spin(7)$
superconformal algebra.
