C₂×#483807 #72287591

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

Aliases

direct product

C₂×#483807

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂²×#509	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216752, C₂×#72216898, C₂×#72216937
maximal normal subgroup of	C₂×#72216752, C₂×#72216898, C₂×#72216937

GAP

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

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

C2×#483807 #72287591