#94841

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=cei,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

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, 2037);