C₂×#151115 #72321886

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#151115

Subgroups

	subgroup	G/H
center, socle	C₂³	#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂²×D₃₂	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

681 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:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225501, #72323079, #72323126
maximal normal subgroup of	C₂×#72225501, #72323079, #72323126
maximal quotient of	#72321894, #72321955

GAP

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

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

C2×#151115 #72321886