C₂×#479958 #72318730

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#479958

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂²×#97
prime radical	C₂⁵	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3329 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)(9,10)(11,12,14,20,16,22,15,34,18,24,33,41,17,23,32,21)(13,25,27,38,29,19,28,37,26,39,31,42,30,36,35,40);(3,4,8,10,6,7,5,9)(11,13,17,30,18,26,16,29)(12,19,22,39,24,36,23,25)(14,31,32,28,33,27,15,35)(20,40,34,38,41,37,21,42);(4,7)(9,10)(13,26)(19,36)(25,39)(27,31)(28,35)(29,30)(37,40)(38,42);(13,26)(19,36)(25,39)(27,31)(28,35)(29,30)(37,40)(38,42)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216937, C₂×#72216701, C₂×#72216715, …
maximal normal subgroup of	C₂×#72216937, C₂×#72216701, C₂×#72216715, …

GAP

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

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

C2×#479958 #72318730