#7625054

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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	#121	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1137 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=i,e^2=fi,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

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, 7532250);