C₂×#7539 #153559

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7539

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂²	#1415
prime radical	C₂×#2262	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 130 points:

(1,2);(3,4,7,14,11,17,10,42)(5,18,20,61,23,64,43,63)(6,24,27,72,31,75,30,90)(8,34,32,92,38,95,37,98)(9,39,33,60,41,100,36,16)(12,44,45,66,49,103,48,19)(13,50,52,86,54,85,102,28)(15,56,55,82,58,81,94,26)(21,68,67,122,69,123,96,114)(22,70,40,62,71,119,101,120)(25,57,76,116,79,118,91,117)(29,87,83,125,89,129,84,74)(35,51,93,111,97,113,99,112)(46,88,104,127,107,130,106,78)(47,108,105,126,109,65,59,77)(53,80,73,110,115,128,124,121);(3,5)(4,12)(6,25)(7,20)(8,35)(9,40)(10,43)(11,23)(13,51)(14,45)(15,57)(16,59)(17,49)(18,24)(19,28)(21,26)(22,36)(27,76)(29,88)(30,91)(31,79)(32,93)(33,71)(34,46)(37,99)(38,97)(39,47)(41,101)(42,48)(44,50)(52,111)(53,114)(54,113)(55,116)(56,67)(58,118)(60,105)(61,72)(62,87)(63,90)(64,75)(65,121)(66,86)(68,73)(69,82)(70,74)(77,80)(78,84)(81,96)(83,127)(85,103)(89,130)(92,104)(94,117)(95,107)(98,106)(100,109)(102,112)(108,110)(115,122)(119,125)(120,129)(123,124)(126,128)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60755);