#7623434

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623323
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3058	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^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,g^c=gh,h^c=hi,e^d=ei

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, 7530630);