Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/Φ(P), where Φ(P) is the Frattini subgroup.
Known results include: The rank of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X.
There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X).
[20] If p is a prime number, then the p-rank of G is the largest rank of an elementary abelian p-subgroup.
[21] The sectional p-rank is the largest rank of an elementary abelian p-section (quotient of a subgroup).