C₂×#483881 #72285877

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#483881

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×#33675	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216771, #72286085, C₂×#72216779, …
maximal normal subgroup of	C₂×#72216771, #72286085, C₂×#72216779, …
maximal quotient of	C₂×#72216334, C₂²×#483881, C₂×#72225053, …

GAP

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

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

C2×#483881 #72285877