C₂×#481339 #72318652

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#481339

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂²×#102
prime radical	C₂×#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5581 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);(4,7,8,9)(11,12,14,20,16,22,31,21,18,24,32,39,17,23,15,33)(13,25,27,34,29,40,42,37,26,38,41,19,30,35,28,36);(3,4,6,8)(5,9,10,7)(11,13)(12,19)(14,28)(15,27)(16,30)(17,29)(18,26)(20,38)(21,40)(22,37)(23,36)(24,34)(25,39)(31,41)(32,42)(33,35);(4,8)(7,9)(13,26)(19,34)(25,38)(27,41)(28,42)(29,30)(35,40)(36,37);(13,26)(19,34)(25,38)(27,41)(28,42)(29,30)(35,40)(36,37)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216916, C₂×#72216937, C₂×#72216707, …
maximal normal subgroup of	C₂×#72216916, C₂×#72216937, C₂×#72216707, …

GAP

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

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

C2×#481339 #72318652