#123

order	32
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #2846 (order 128) · Inn(G) = C₂³ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂³
prime radical	C₂×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

39 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 12 points:

(2,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12);(1,2)(3,4)(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^5|c^2=d,b^a=bd,c^a=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 30);