C₂×#496469 #72321930

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#496469

Subgroups

	subgroup	G/H
center	C₂×C₈	#535
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	#21851
prime radical	C₂×#3444	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(3,4,7,12,8,14,22,29,9,15,23,11,21,13,6,20)(5,10,18,25,19,28,30,34,16,26,31,24,32,27,17,33);(5,16)(17,30)(18,31)(19,32);(1,2);(3,5)(4,10)(6,17)(7,18)(8,19)(9,16)(11,24)(12,25)(13,27)(14,28)(15,26)(20,33)(21,32)(22,30)(23,31)(29,34);(3,6,21,23,9,22,8,7)(4,11,14,20,15,12,13,29)(5,17,32,31,16,30,19,18)(10,24,28,33,26,25,27,34)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72217274, C₂×#72224957, C₂×#72224961
maximal normal subgroup of	C₂×#72217274, C₂×#72224957, C₂×#72224961

GAP

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

gap> G := Group((3,4,7,12,8,14,22,29,9,15,23,11,21,13,6,20)(5,10,18,25,19,28,30,34,16,26,31,24,32,27,17,33), (5,16)(17,30)(18,31)(19,32), (1,2), (3,5)(4,10)(6,17)(7,18)(8,19)(9,16)(11,24)(12,25)(13,27)(14,28)(15,26)(20,33)(21,32)(22,30)(23,31)(29,34), (3,6,21,23,9,22,8,7)(4,11,14,20,15,12,13,29)(5,17,32,31,16,30,19,18)(10,24,28,33,26,25,27,34));

C2×#496469 #72321930