Double veronese cones with 28 nodes

We study nodal del Pezzo 3-folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all G-birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group S4 into the space Cremona group.