#489664

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

624 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,24,30)(19,31,36,63)(20,26,40,56)(21,28,43,58)(22,27,42,59)(23,46,45,29)(25,48,52,34)(32,62,67,79)(33,51,44,61)(35,50,47,60)(37,69,41,54)(38,57,72,53)(39,73,71,55)(49,70,77,65)(64,80,68,76)(66,78,74,75);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,24,36)(18,25,30,52)(20,32,40,67)(21,34,43,48)(22,44,42,33)(23,47,45,35)(26,49,56,77)(27,31,59,63)(28,60,58,50)(29,61,46,51)(37,70,41,65)(38,64,72,68)(39,74,71,66)(53,79,57,62)(54,75,69,78)(55,80,73,76);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20)(18,26)(19,32)(21,41)(22,38)(23,39)(24,40)(25,49)(27,57)(28,54)(29,55)(30,56)(31,62)(33,68)(34,65)(35,66)(36,67)(37,43)(42,72)(44,64)(45,71)(46,73)(47,74)(48,70)(50,78)(51,76)(52,77)(53,59)(58,69)(60,75)(61,80)(63,79);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,42,24,43,45,22)(18,27,29,58,30,59,46,28)(19,33,35,34,36,44,47,48)(20,37,39,38,40,41,71,72)(25,50,51,31,52,60,61,63)(26,53,55,54,56,57,73,69)(32,64,66,70,67,68,74,65)(49,75,76,79,77,78,80,62)

Polycyclic presentation:

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

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, 396860);