We introduce the first examples of groups \(G\) with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, \(\mathcal{L}(G)\). Specifically, assume that \(G=A\times W\), where \(A\) is an infinite abelian group and \(W\) is an ICC wreath-like product group CIOS22a; AMCOS23 with property (T) and trivial abelianization. Then whenever \(H\) is an \emph{arbitrary} group such that \(\mathcal{L}(G)\) is \(\ast\)-isomorphic to \(\mathcal L(H)\), via an \emph{arbitrary} \(\ast\)-isomorphism preserving the canonical traces, it must be the case that \(H= B \times H_0\) where \(B\) is infinite abelian and \(H_0\) is isomorphic to \(W\). Moreover, we completely describe the \(\ast\)-isomorphism between \(\mathcal L(G)\) and \(\mathcal L(H)\). This yields new applications to the classification of group C\(^*\)-algebras, including examples of non-amenable groups which are recoverable from their reduced C\(^*\)-algebras but not from their von Neumann algebras.