#489612

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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	#62341	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

671 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,24,30)(19,31,35,63)(20,26,40,56)(21,28,42,57)(22,27,41,58)(23,46,45,29)(25,48,51,34)(32,64,67,62)(33,52,43,61)(36,50,47,59)(37,60,70,54)(38,71,44,53)(39,73,72,55)(49,69,77,66)(65,79,75,78)(68,80,74,76);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19)(18,25)(20,32)(21,34)(22,43)(23,47)(24,35)(26,49)(27,31)(28,59)(29,61)(30,51)(33,41)(36,45)(37,69)(38,65)(39,74)(40,67)(42,48)(44,75)(46,52)(50,57)(53,62)(54,76)(55,79)(56,77)(58,63)(60,80)(64,71)(66,70)(68,72)(73,78);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20,24,40)(18,26,30,56)(19,32,35,67)(21,37,42,70)(22,44,41,38)(23,39,45,72)(25,49,51,77)(27,53,58,71)(28,60,57,54)(29,55,46,73)(31,62,63,64)(33,65,43,75)(34,69,48,66)(36,68,47,74)(50,76,59,80)(52,78,61,79);(2,9)(3,12)(8,10)(13,14)(17,21,23,41,24,42,45,22)(18,27,29,57,30,58,46,28)(19,33,36,34,35,43,47,48)(20,37,39,38,40,70,72,44)(25,50,52,31,51,59,61,63)(26,53,55,54,56,71,73,60)(32,65,68,69,67,75,74,66)(49,76,78,62,77,80,79,64)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396808);