C₂×#513082 #72231995

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#513082

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁵
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 42 points:

(1,2);(3,4,6,7)(5,8,9,10)(12,17)(13,22)(14,23)(15,26)(18,28)(19,25)(20,36)(21,37)(24,35)(27,40)(29,30)(31,33)(32,42)(34,41)(38,39);(4,7)(8,10)(12,18)(17,28)(20,32)(21,33)(29,41)(30,34)(31,37)(36,42);(11,12,19,34,14,20,24,31,26,33,39,36,40,41,13,17,16,18,22,30,27,32,38,37,15,21,35,42,23,29,25,28);(3,5,6,9)(4,8,7,10)(12,18)(17,28)(20,32)(21,33)(29,41)(30,34)(31,37)(36,42)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216311
maximal normal subgroup of	C₂×#72216311

GAP

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

gap> G := Group((1,2), (3,4,6,7)(5,8,9,10)(12,17)(13,22)(14,23)(15,26)(18,28)(19,25)(20,36)(21,37)(24,35)(27,40)(29,30)(31,33)(32,42)(34,41)(38,39), (4,7)(8,10)(12,18)(17,28)(20,32)(21,33)(29,41)(30,34)(31,37)(36,42), (11,12,19,34,14,20,24,31,26,33,39,36,40,41,13,17,16,18,22,30,27,32,38,37,15,21,35,42,23,29,25,28), (3,5,6,9)(4,8,7,10)(12,18)(17,28)(20,32)(21,33)(29,41)(30,34)(31,37)(36,42));

C2×#513082 #72231995