#488851

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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	#466	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396047);