#488594

order	512
nilpotency class	4
isoclinism	isoclinic class of #488549
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

681 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,24,30)(19,31,35,59)(20,36,37,56)(21,40,44,54)(22,29,23,28)(25,47,50,33)(26,45,51,39)(27,42,55,43)(32,60,61,77)(34,65,46,53)(38,69,68,52)(41,57,72,49)(48,70,75,80)(58,64,79,63)(62,66,74,76)(67,73,78,71);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,24,35)(18,25,30,50)(20,32,37,61)(21,41,44,72)(22,34,23,46)(26,48,51,75)(27,53,55,65)(28,49,29,57)(31,54,59,40)(33,42,47,43)(36,66,56,76)(38,70,68,80)(39,63,45,64)(52,77,69,60)(58,78,79,67)(62,73,74,71);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,20)(18,26)(19,32)(21,38)(22,45)(23,39)(24,37)(25,48)(27,52)(28,56)(29,36)(30,51)(31,58)(33,62)(34,64)(35,61)(40,67)(41,70)(42,73)(43,71)(44,68)(46,63)(47,74)(49,76)(50,75)(53,77)(54,78)(55,69)(57,66)(59,79)(60,65)(72,80);(2,8)(6,13)(7,14)(12,15)(20,37)(26,51)(32,61)(36,56)(38,68)(39,45)(48,75)(52,69)(58,79)(60,77)(62,74)(63,64)(66,76)(67,78)(70,80)(71,73)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395790);