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