#332638

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 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)(3,8)(4,14)(5,15)(6,16)(7,11)(9,12)(10,13)(17,18,24,30)(19,31,35,50)(20,26,40,54)(21,41,44,57)(22,29,23,28)(25,36,49,34)(27,45,56,43)(32,60,62,51)(33,63,64,48)(37,67,66,47)(38,69,72,76)(39,58,46,53)(42,55,65,59)(52,71,77,70)(61,74,79,80)(68,78,73,75);(1,3,7,6)(2,8,11,16)(4,12,5,13)(9,15,10,14)(17,19)(18,25)(20,37)(21,42)(22,36)(23,34)(24,35)(26,51)(27,55)(28,50)(29,31)(30,49)(32,39)(33,43)(38,61)(40,66)(41,48)(44,65)(45,64)(46,62)(47,53)(52,74)(54,60)(56,59)(57,63)(58,67)(68,70)(69,75)(71,73)(72,79)(76,78)(77,80);(1,4,7,5)(2,9,11,10)(3,12,6,13)(8,15,16,14)(17,20)(18,26)(19,32)(21,38)(22,39)(23,46)(24,40)(25,47)(27,52)(28,53)(29,58)(30,54)(31,51)(33,61)(34,37)(35,62)(36,66)(41,69)(42,73)(43,70)(44,72)(45,71)(48,74)(49,67)(50,60)(55,78)(56,77)(57,76)(59,75)(63,80)(64,79)(65,68);(1,5,7,4)(2,10,11,9)(3,13,6,12)(8,14,16,15)(17,21,24,44)(18,27,30,56)(19,33,35,64)(20,38,40,72)(22,45,23,43)(25,48,49,63)(26,52,54,77)(28,57,29,41)(31,59,50,55)(32,61,62,79)(34,65,36,42)(37,68,66,73)(39,71,46,70)(47,74,67,80)(51,75,60,78)(53,76,58,69)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239834);