#94832

order	512
nilpotency class	7
isoclinism	isoclinic class of D₁₂₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₆₄
commutator	C₆₄	C₂×C₄
Frattini, cosocle	C₂×C₆₄	C₂²
prime radical	C₂×D₆₄	C₂
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

57 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,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, 2028);