#7196828

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=ghi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104024);