TITLE: The smallest group with 3 generators
QUESTION [10 upvotes]: What is the smallest (in terms of the number of elements) nonabelian group such that any presentation requires at least 3 generators? Most of the nonabelian finite groups I know seem to require only 2 generators.

REPLY [13 votes]: Without having thought about it for long, I would guess that the smallest examples
have order 16. For example $D_8 \times C_2$ or $Q_8 \times C_2$. They have elementary abelian quotients of order 8, so they definitely need three generators.