#263572

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

758 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)(5,8)(6,11)(7,13)(10,16)(14,15)(17,18,24,30)(19,31,35,55)(20,26,39,52)(21,40,43,53)(22,29,23,28)(25,44,48,34)(27,42,54,41)(32,49,61,57)(33,62,64,47)(36,46,60,72)(37,66,68,76)(38,56,45,51)(50,70,74,67)(58,69,75,63)(59,71,79,80)(65,77,78,73);(4,13)(6,15)(7,12)(11,14)(17,19,23,44,24,35,22,34)(18,25,29,55,30,48,28,31)(20,36,45,61,39,60,38,32)(21,33,42,69,43,64,41,63)(26,49,56,72,52,57,51,46)(27,47,53,75,54,62,40,58)(37,65,70,79,68,78,67,59)(50,73,76,80,74,77,66,71);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,20)(18,26)(19,32)(21,37)(22,38)(23,45)(24,39)(25,46)(27,50)(28,51)(29,56)(30,52)(31,57)(33,59)(34,60)(35,61)(36,44)(40,66)(41,67)(42,70)(43,68)(47,71)(48,72)(49,55)(53,76)(54,74)(58,77)(62,80)(63,78)(64,79)(65,69)(73,75);(3,10)(6,15)(9,16)(11,14)(17,21,23,42,24,43,22,41)(18,27,29,53,30,54,28,40)(19,33,44,69,35,64,34,63)(20,37,45,70,39,68,38,67)(25,47,55,75,48,62,31,58)(26,50,56,76,52,74,51,66)(32,59,36,65,61,79,60,78)(46,71,49,73,72,80,57,77)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170768);