2-local nilpotency class

Definition
Suppose $$G$$ is a group. The 2-local nilpotency class is defined as the 2-defining ingredient::local nilpotency class of $$G$$. Explicitly, it is the supremum, over 2-generated subgroups $$H$$ of $$G$$, of the defining ingredient::nilpotency class of $$H$$. In other words, it is defined as:

$$\sup_{x,y \in G} \operatorname{class}(\langle x,y \rangle)$$

(Note that $$x,y$$ are allowed to be equal to each other, but this does not matter for nontrivial groups).

If there is a non-nilpotent subgroup of $$G$$ generated by two elements, then $$G$$ is not 2-locally nilpotent. It is also possible that $$G$$ be non-nilpotent because, while each 2-generated subgroup is nilpotent, there is no upper bound on the nilpotency class. An example is the generalized dihedral group for 2-quasicyclic group.

In general, the 2-local nilpotency class of a nilpotent group is less than or equal to its nilpotency class.