3-central implies 9-abelian

Statement
Suppose $$G$$ is a 2-central group: its inner automorphism group is a group of exponent three, or equivalently, every cube element of $$G$$ is in the center.

Then, $$G$$ is a 9-abelian group: the ninth power map is an endomorphism, and hence a universal power endomorphism, of $$G$$. In symbols:

$$(xy)^9 = x^9y^9 \ \forall \ x,y \in G$$

Related facts

 * 2-central implies 4-abelian
 * 4-central implies 16-abelian
 * 6-central implies 36-abelian