#94612

order	512
nilpotency class	6
isoclinism	isoclinic class of #7540
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1415
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	#7383
prime radical	#2262	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

73 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230739
maximal normal subgroup of	#72230739

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1808);