#94697

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	SD₆₄
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

86 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fgi,d^2=i,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, 1893);