#488726

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
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	#62279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

712 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)(17,18,25,32)(19,33,38,65)(20,39,43,62)(21,29,46,59)(22,28,45,60)(23,31,24,30)(26,50,54,36)(27,48,57,42)(34,66,71,77)(35,53,47,63)(37,52,49,61)(40,55,44,58)(41,75,76,56)(51,67,79,72)(64,70,80,69)(68,73,74,78);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,25,38)(18,26,32,54)(20,34,43,71)(21,36,46,50)(22,47,45,35)(23,37,24,49)(27,51,57,79)(28,33,60,65)(29,61,59,52)(30,53,31,63)(39,73,62,78)(40,74,44,68)(41,67,76,72)(42,69,48,70)(55,80,58,64)(56,77,75,66);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,27)(19,34)(21,44)(22,41)(23,48)(24,42)(25,43)(26,51)(28,58)(29,56)(30,62)(31,39)(32,57)(33,64)(35,72)(36,68)(37,70)(38,71)(40,46)(45,76)(47,67)(49,69)(50,74)(52,66)(53,78)(54,79)(55,60)(59,75)(61,77)(63,73)(65,80);(2,8)(6,13)(7,14)(12,15)(17,21,24,45,25,46,23,22)(18,28,31,59,32,60,30,29)(19,35,37,36,38,47,49,50)(20,40,42,41,43,44,48,76)(26,52,53,33,54,61,63,65)(27,55,39,56,57,58,62,75)(34,67,70,74,71,72,69,68)(51,77,78,80,79,66,73,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395922);