#347033

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254229);