A torsion-free group of nilpotency class six with all subgroups subnormal of defect at most $5$

Abstract

Casolo asked whether a torsion-free group in which every subgroup is subnormal of defect at most $n$ must be nilpotent of class at most $n$. The answer is known to be positive for $n\leq 4$. We construct a $2$-generated torsion-free nilpotent group $G$ such that $$ \cl(G)=6 \qquad\text{and}\qquad [G,{}_5H]\leq H \quad\text{for every }H\leq G. $$ Thus Casolo's question has a negative answer for $n=5$.