#7624409

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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	#2186	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1158 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)(17,18,22,28)(19,31,33,25)(20,36,38,70)(21,30,24,27)(23,43,42,54)(26,50,52,39)(29,41,55,45)(32,60,61,69)(34,48,40,53)(35,67,65,74)(37,71,46,56)(44,57,73,51)(47,63,58,78)(49,66,59,72)(62,79,64,77)(68,76,80,75);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,33)(18,25,28,31)(20,32,38,61)(21,40,24,34)(23,35,42,65)(26,47,52,58)(27,53,30,48)(29,49,55,59)(36,69,70,60)(37,64,46,62)(39,63,50,78)(41,72,45,66)(43,74,54,67)(44,68,73,80)(51,76,57,75)(56,77,71,79);(17,20)(18,26)(19,32)(21,37)(22,38)(23,44)(24,46)(25,47)(27,51)(28,52)(29,56)(30,57)(31,58)(33,61)(34,62)(35,68)(36,43)(39,45)(40,64)(41,50)(42,73)(48,75)(49,77)(53,76)(54,70)(55,71)(59,79)(60,67)(63,66)(65,80)(69,74)(72,78);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,33)(25,31)(32,61)(34,40)(35,65)(47,58)(48,53)(49,59)(60,69)(62,64)(63,78)(66,72)(67,74)(68,80)(75,76)(77,79);(2,8)(6,9)(7,16)(14,15)(17,21,22,24)(18,27,28,30)(19,34,33,40)(20,37,38,46)(23,41,42,45)(25,48,31,53)(26,51,52,57)(29,54,55,43)(32,62,61,64)(35,66,65,72)(36,56,70,71)(39,44,50,73)(47,75,58,76)(49,74,59,67)(60,77,69,79)(63,80,78,68)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531605);