#513144

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	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

294 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)(19,30)(20,26)(21,25)(22,41)(23,42)(24,31)(27,38)(28,39)(29,37)(32,43)(33,65)(34,46)(35,48)(36,70)(40,54)(44,61)(45,66)(47,58)(49,71)(50,69)(51,67)(52,68)(53,72)(55,63)(56,62)(57,60)(59,73)(64,76)(74,80)(75,78)(77,79);(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)(23,34)(25,47)(26,30)(27,44)(28,45)(29,46)(35,58)(36,59)(37,60)(38,63)(39,62)(40,64)(41,73)(42,57)(43,48)(49,77)(50,76)(51,55)(52,56)(53,74)(54,75)(61,67)(65,70)(66,68)(69,78)(71,80)(72,79);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,31)(21,38)(22,39)(23,37)(24,43)(26,48)(27,28)(29,54)(30,47)(32,61)(33,66)(35,67)(36,68)(40,72)(41,49)(42,50)(44,56)(45,55)(46,76)(51,52)(53,70)(57,75)(58,63)(59,62)(64,80)(65,77)(69,71)(73,74)(78,79);(2,8)(6,9)(7,16)(14,15)(18,26)(19,31)(25,48)(27,51)(28,52)(29,42)(32,58)(33,59)(34,60)(41,70)(49,53)(50,54)(61,63)(62,66)(64,78)(79,80)

Polycyclic presentation:

2^9|b^2=ei,c^2=ei,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, 420340);