C₂×#151055 #72231998

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

Aliases

direct product

C₂×#151055

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#367
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂⁴	#367
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

679 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216311
maximal normal subgroup of	C₂×#72216311

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

C2×#151055 #72231998