#94695

order	512
nilpotency class	6
isoclinism	isoclinic class of #94694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7418
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂×M₆₄	C₂²
prime radical	C₂³	D₃₂
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fgi,e^2=ghi,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fgi,g^a=gh,h^a=hi,c^b=cf,d^b=defgi,e^b=egi,f^b=fgi,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1891);