#7623512

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	#33769	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530708);