  Motivated by recent results on the minimal base of a permutation group, we
introduce a new local invariant attached to arbitrary finite groups. More
precisely, a subset Delta of a finite group G is called a p-base (where p is a
prime) if Delta generates a p-group and C_G(Delta) is p-nilpotent. Building on
results of Halasi-Mar\'oti, we prove that p-solvable groups possess p-bases of
size 3 for every prime p. For other prominent groups we exhibit p-bases of size
2. In fact, we conjecture the existence of p-bases of size 2 for every finite
group. Finally, the notion of p-bases is generalized to blocks and fusion
systems.
