#490126

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62437	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397322);