#489346

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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	#61502	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

675 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)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,23,28)(19,29,31,58)(20,25,34,49)(21,37,42,72)(22,43,40,27)(24,45,47,32)(26,52,55,41)(30,59,61,57)(33,64,44,53)(35,66,39,71)(36,69,65,51)(38,56,74,48)(46,63,76,60)(50,75,54,68)(62,80,70,78)(67,79,73,77);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,30)(21,38)(22,44)(23,31)(25,46)(26,53)(27,56)(28,47)(29,37)(32,41)(33,40)(34,61)(35,67)(36,70)(39,73)(42,74)(43,48)(45,52)(49,76)(50,78)(51,79)(54,80)(55,64)(57,71)(58,72)(59,66)(60,68)(62,65)(63,75)(69,77);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20,23,34)(18,25,28,49)(19,30,31,61)(21,39,42,35)(22,36,40,65)(24,46,47,76)(26,54,55,50)(27,51,43,69)(29,57,58,59)(32,63,45,60)(33,62,44,70)(37,71,72,66)(38,73,74,67)(41,75,52,68)(48,77,56,79)(53,80,64,78);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,31)(20,34)(24,47)(25,49)(29,58)(32,45)(33,44)(35,39)(36,65)(38,74)(48,56)(50,54)(51,69)(53,64)(66,71)(68,75)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396542);