#6482203

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 80 points:

(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,24,29)(19,30,35,55)(20,36,40,26)(21,27,42,52)(22,28,23,44)(25,33,47,34)(31,56,60,53)(32,54,62,71)(37,65,68,48)(38,66,39,49)(41,50,43,51)(45,59,64,58)(46,63,72,61)(57,76,78,75)(67,74,69,73)(70,77,80,79);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,33,24,35,23,34)(18,25,44,55,29,47,28,30)(20,31,38,58,40,60,39,59)(21,32,43,63,42,62,41,61)(26,45,66,56,36,64,49,53)(27,46,50,71,52,72,51,54)(37,57,69,79,68,78,67,77)(48,70,73,76,65,80,74,75);(3,9)(6,13)(10,11)(14,15)(17,20,23,39,24,40,22,38)(18,26,28,49,29,36,44,66)(19,31,34,59,35,60,33,58)(21,37,41,67,42,68,43,69)(25,45,30,53,47,64,55,56)(27,48,51,74,52,65,50,73)(32,57,61,77,62,78,63,79)(46,70,54,75,72,80,71,76);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21)(18,27)(19,32)(20,37)(22,43)(23,41)(24,42)(25,46)(26,48)(28,51)(29,52)(30,54)(31,57)(33,63)(34,61)(35,62)(36,65)(38,69)(39,67)(40,68)(44,50)(45,70)(47,72)(49,74)(53,75)(55,71)(56,76)(58,79)(59,77)(60,78)(64,80)(66,73);(2,8)(6,13)(7,16)(14,15)(17,22,24,23)(18,28,29,44)(19,33,35,34)(20,38,40,39)(21,41,42,43)(25,30,47,55)(26,49,36,66)(27,50,52,51)(31,58,60,59)(32,61,62,63)(37,67,68,69)(45,53,64,56)(46,71,72,54)(48,73,65,74)(57,77,78,79)(70,76,80,75)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389399);