#484978

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
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	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

697 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392174);