SmallGroup(32,2)

Definition
This group can be defined by the following presentation:

$$G := \langle a_1,a_2,a_3 \mid a_1^4 = a_2^4 = a_3^2 = e, [a_1,a_2] = a_3, [a_1,a_3] = [a_2,a_3] = e \rangle$$

where $$e$$ denotes the identity element and $$[, ]$$ stands for the commutator of two elements (the isomorphism type of the group is independent of the choice of commutator map).

See the section for how to construct the group using this presentation in GAP.

Description by presentation
gap> F := FreeGroup(3);  gap> G := F/[F.1^4,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)];  gap> IdGroup(G); [ 32, 2 ]

GAP verification of function values and group properties
Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set G := SmallGroup(32,2); or set it to this group in any other manner:

gap> Order(G); 32 gap> Exponent(G); 4 gap> NilpotencyClassOfGroup(G); 2 gap> DerivedLength(G); 2 gap> FrattiniLength(G); 2 gap> Rank(G); 2 gap> RankAsPGroup(G); 3 gap> NormalRank(G); 3 gap> CharacteristicRank(G); 3