#490357

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	#62383	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

652 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,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, 397553);