C₂×#150158 #72322807

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#150158

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#33733	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 42 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322835, #72323020, #72323113
maximal normal subgroup of	#72322835, #72323020, #72323113
maximal quotient of	#72322819

GAP

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

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

C2×#150158 #72322807