C₂×#122474 #72322367

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#122474

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂³	C₂×D₄⋊C₈
prime radical	C₂³×D₈	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72286692, C₂×#72286314, C₂×#72286654, …
maximal normal subgroup of	C₂×#72286692, C₂×#72286314, C₂×#72286654, …
maximal quotient of	#72322441, #72322503, C₂×#72286339

GAP

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

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

C2×#122474 #72322367