Connes Rigidity Conjecture (1980) states that any ICC (infinite conjugacy class) property (T) group is W*-superrigid, meaning the group can be completely recognized from its group von Neumann algebra. The first examples of groups satisfying the conjecture, wreath-like product groups, were constructed in the work of Chifan-Ioana-Osin-Sun (2021). Building on these groups, we investigate the reconstruction of groups with infinite center from their group von Neumann algebras. We introduce the first examples of groups with infinite center whose direct product structure and ICC part are completely recognizable, and the first examples of property (T) W*-superrigid groups with infinite center.  This is based on joint work with Ionuţ Chifan and Adriana Fernández Quero, and upcoming joint work with Ionuţ Chifan, Adriana Fernández Quero and Denis Osin.