C₂²×#13174 #72313628

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#13174

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(6,8)(10,14,16,23)(11,18)(12,21)(15,22,24,17)(19,20);(1,2);(3,4);(1,2)(3,4)(5,6,7,8)(10,15)(11,19)(12,21)(14,22)(16,24)(17,23)(18,20);(1,2)(3,4)(9,10,11,14,12,17,18,24,13,16,20,23,21,22,19,15)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#498375, C₂²×#497864, C₂²×#495895
maximal normal subgroup of	C₂²×#498375, C₂²×#497864, C₂²×#495895

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,8)(10,14,16,23)(11,18)(12,21)(15,22,24,17)(19,20), (1,2), (3,4), (1,2)(3,4)(5,6,7,8)(10,15)(11,19)(12,21)(14,22)(16,24)(17,23)(18,20), (1,2)(3,4)(9,10,11,14,12,17,18,24,13,16,20,23,21,22,19,15));

C22×#13174 #72313628