The collective electronic oscillators method is used to construct an effective Frenkel exciton Hamiltonian for conjugated dendrimers with fractal geometry. Self-similarity and the high degree of symmetry utilized by decomposing the space of optical excitations into irreducible representations make it possible to compute the one-exciton states and the linear optical response with reduced numerical effort that scales linearly rather than exponentially with the number of generations. The linear optical response is dominated by localized excitons belonging to the periphery.