#41×#360 #72342526

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

Aliases

direct product

#41×#360

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#102
prime radical	#41×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#360, #41×AGL(1,ℤ/16), #72342658
maximal normal subgroup of	AGL(1,ℤ/8)×#360, #41×AGL(1,ℤ/16), #72342658

GAP

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

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