The exact, primary response of a nonlinear structural system with symmetry can be obtained with a reduced number of degrees of freedom or symmetry modes [the second author, A group theoretic approach to computational bifurcation problems with symmetry, Comput. Methods Appl. Mech. Eng. (to appear)]. A group-theoretic approach shows that the set of symmetry transformations of the undeformed structure can be used to construct a projection matrix from configuration space onto the symmetry subspace spanned by the symmetry modes. A numerical approach to the computation of symmetry modes is presented in this paper. The technique exploits the fact that the symmetry modes are the eigenvectors of the projection matrix corresponding to a repeated eigenvalue of unity. The multiplicity of the unit eigenvalue is equal to the trace of the projection matrix. Thus, the symmetry modes can be extracted by inverse iteration combined with matrix deflation. Several examples are presented to illustrate the method.