#7624950

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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	#2218	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1149 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)(17,18,24,29)(19,25,33,49)(20,34,36,61)(21,27,23,44)(22,42,43,69)(26,50,51,38)(28,41,53,35)(30,54,56,76)(31,46,40,48)(32,59,60,74)(37,62,39,67)(45,70,71,58)(47,68,75,55)(52,64,63,66)(57,77,65,78)(72,80,73,79);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,30)(21,40)(22,32)(23,31)(24,33)(26,45)(27,48)(28,47)(29,49)(34,54)(35,55)(36,56)(37,65)(38,58)(39,57)(41,68)(42,59)(43,60)(44,46)(50,70)(51,71)(52,73)(53,75)(61,76)(62,78)(63,72)(64,79)(66,80)(67,77)(69,74);(17,20,35,64,23,39,43,50,24,36,41,66,21,37,22,38)(18,26,42,62,27,52,53,61,29,51,69,67,44,63,28,34)(19,30,55,79,31,57,60,70,33,56,68,80,40,65,32,58)(25,45,59,78,48,73,75,76,49,71,74,77,46,72,47,54);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,36)(26,51)(30,56)(34,61)(37,39)(38,50)(45,71)(52,63)(54,76)(57,65)(58,70)(62,67)(64,66)(72,73)(77,78)(79,80);(2,8)(6,9)(7,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,33,40)(20,37,36,39)(22,41,43,35)(25,46,49,48)(26,52,51,63)(28,42,53,69)(30,57,56,65)(32,55,60,68)(34,62,61,67)(38,66,50,64)(45,72,71,73)(47,74,75,59)(54,77,76,78)(58,79,70,80)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532146);