Symmetric cubature formulas for a truncated octahedron

For a truncated octahedron, which can be used to fill the whole space $\mathbf{R}^3$ by translating it, we construct symmetric cubature formulas, exact for polynomials of degrees 3, 5, and 7. We furnish estimates of the remainder terms, and we discuss the problem of numerical integration over an arbitrary bounded domain $D\subset\mathbf{R}^3$.