#7623446

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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	#3061	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1205 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,29)(19,24,31,43)(20,26,34,49)(21,25,39,48)(22,40,38,28)(27,52,47,37)(30,42,55,69)(32,41,56,65)(33,59,57,45)(35,46,60,71)(36,63,61,51)(44,70,66,58)(50,73,72,62)(53,64,76,79)(54,77,75,68)(67,80,78,74);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,31)(18,24,29,43)(20,30,34,55)(21,32,39,56)(22,33,38,57)(25,41,48,65)(26,42,49,69)(27,44,47,66)(28,45,40,59)(35,53,60,76)(36,54,61,75)(37,58,52,70)(46,64,71,79)(50,67,72,78)(51,68,63,77)(62,74,73,80);(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)(32,58)(33,57)(35,62)(36,61)(39,52)(40,47)(42,64)(43,65)(44,45)(49,71)(50,51)(53,74)(54,75)(56,70)(59,66)(60,73)(63,72)(67,68)(69,79)(76,80)(77,78);(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)(31,55)(32,53)(33,54)(37,62)(38,61)(39,60)(40,63)(41,64)(43,69)(44,67)(45,68)(47,72)(48,71)(52,73)(56,76)(57,75)(58,74)(59,77)(65,79)(66,78)(70,80);(19,31)(20,34)(24,43)(26,49)(32,56)(33,57)(35,60)(36,61)(41,65)(44,66)(45,59)(46,71)(50,72)(51,63)(58,70)(62,73)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530642);