#94840

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

Polycyclic presentation:

2^9|a^2=d,b^2=ce,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, 2036);