#490190

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
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	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

614 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)(3,12)(4,13)(5,9)(6,10)(7,11)(8,15)(14,16)(17,18,23,28)(19,29,32,58)(20,25,37,52)(21,38,42,74)(22,43,40,27)(24,45,47,31)(26,53,55,41)(30,59,62,57)(33,64,44,54)(34,65,35,69)(36,72,67,51)(39,56,75,48)(46,60,76,61)(49,71,50,68)(63,78,73,79)(66,77,70,80);(1,3,7,8)(2,4,11,16)(5,14,6,13)(9,12,10,15)(17,19)(18,24)(20,30)(21,39)(22,44)(23,32)(25,46)(26,54)(27,56)(28,47)(29,38)(31,41)(33,40)(34,66)(35,70)(36,73)(37,62)(42,75)(43,48)(45,53)(49,78)(50,79)(51,80)(52,76)(55,64)(57,65)(58,74)(59,69)(60,68)(61,71)(63,67)(72,77);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,20)(18,25)(19,30)(21,34)(22,36)(23,37)(24,46)(26,49)(27,51)(28,52)(29,57)(31,60)(32,62)(33,63)(35,42)(38,65)(39,66)(40,67)(41,68)(43,72)(44,73)(45,61)(47,76)(48,77)(50,55)(53,71)(54,78)(56,80)(58,59)(64,79)(69,74)(70,75);(1,5,7,6)(2,9,11,10)(3,13,8,14)(4,15,16,12)(17,21,40,53,23,42,22,41)(18,26,43,74,28,55,27,38)(19,31,44,75,32,45,33,39)(20,34,67,71,37,35,36,68)(24,29,56,64,47,58,48,54)(25,49,72,69,52,50,51,65)(30,60,73,70,62,61,63,66)(46,57,80,79,76,59,77,78)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397386);