Exponent five and locally nilpotent not implies nilpotent

Statement
It is possible to have a group $$G$$ such that:


 * 1) The exponent of $$G$$ equals 5. In particular, $$G$$ is a group of prime exponent.
 * 2) $$G$$ is a locally nilpotent group.
 * 3) $$G$$ is not a nilpotent group. In fact, we can choose $$G$$ so as not to be a solvable group.

Related facts

 * Exponent two implies abelian
 * Exponent three implies class three