#94833

order	512
nilpotency class	7
isoclinism	isoclinic class of D₁₂₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	D₆₄
commutator	C₆₄	C₂×C₄
Frattini, cosocle	C₂×C₆₄	C₂²
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

53 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 132 points:

(1,2,3,4)(6,13)(7,19)(8,21)(9,27)(10,32)(11,36)(14,44)(15,49)(16,53)(17,56)(18,43)(20,26)(22,67)(23,72)(24,76)(25,39)(28,84)(29,88)(30,45)(31,40)(33,93)(34,50)(35,41)(37,54)(38,42)(46,109)(47,112)(48,106)(51,116)(52,115)(55,117)(57,58)(59,69)(60,74)(61,85)(62,89)(63,81)(64,65)(66,86)(68,78)(70,126)(71,100)(73,79)(75,103)(77,80)(82,83)(87,110)(90,114)(91,113)(92,102)(94,101)(95,104)(96,105)(97,98)(99,118)(107,108)(111,127)(119,124)(120,123)(121,129)(122,132)(125,130)(128,131);(5,6,12,18)(7,20,25,63)(8,26,30,81)(9,31,34,92)(10,35,37,95)(11,38,36,17)(13,39,43,19)(14,45,48,21)(15,50,52,27)(16,54,55,32)(22,68,71,125)(23,73,75,128)(24,77,74,62)(28,85,87,132)(29,89,86,80)(33,94,82,91)(40,100,102,67)(41,103,104,72)(42,60,56,76)(44,97,106,57)(46,110,111,84)(47,66,96,88)(49,98,115,58)(51,83,90,93)(53,99,117,59)(61,118,122,69)(64,79,107,131)(65,78,108,130)(70,127,119,109)(101,124,113,126)(105,120,112,129)(114,121,116,123)

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=ce,e^b=ef,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2029);