#347467

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
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	#63067	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254663);