#513141

order	512
nilpotency class	5
isoclinism	isoclinic class of #513098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33740	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,d^b=di,f^c=fg,g^c=gh,h^c=hi

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, 420337);