#513142

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	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

288 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)(17,18,23,29)(19,24,30,43)(20,25,37,49)(21,38,40,67)(22,41,39,28)(26,50,52,34)(27,53,54,35)(31,42,56,68)(32,62,57,78)(33,63,58,46)(36,55,64,48)(44,73,71,59)(45,75,72,60)(47,66,51,65)(61,76,77,70)(69,80,74,79);(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)(23,30)(25,42)(26,44)(27,45)(28,46)(29,43)(34,59)(35,60)(36,61)(37,56)(38,62)(39,58)(40,57)(41,63)(47,69)(48,70)(49,68)(50,73)(51,74)(52,71)(53,75)(54,72)(55,76)(64,77)(65,79)(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)(28,55)(29,49)(31,59)(32,60)(33,58)(36,66)(37,50)(38,47)(40,53)(41,48)(43,68)(44,45)(46,76)(51,67)(52,54)(56,73)(57,75)(61,80)(62,69)(63,70)(64,65)(71,72)(74,78)(77,79);(2,8)(6,9)(7,16)(14,15)(19,30)(24,43)(31,56)(32,57)(33,58)(42,68)(44,71)(45,72)(46,63)(59,73)(60,75)(61,77)(62,78)(69,74)(70,76)(79,80)

Polycyclic presentation:

2^9|a^2=i,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, 420338);