© 2021, The Author(s), under exclusive licence to Springer Nature B.V.We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for a one-dimensional lattice model considering higher order effects, and later it was re-derived in the context of nonlinear nonlocal elasticity. In the sense of one-dimensional wave propagation in solids, the nonlinearity function of the equation is connected with the stress-strain relation of the physical model. Considering a general nonlinearity, we determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. We find that the maximal dimension of the symmetry algebra is four, which is realized when the nonlinearity assumes some special canonical form. After that we perform reductions to ordinary differential equations. In case of a quadratic nonlinearity, we provide several exact solutions, some of which are in terms of elliptic functions.