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 $\mathfrak{S}_4$ into the space Cremona group.