#347036

order	512
nilpotency class	3
isoclinism	isoclinic class of #347034
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=h,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=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254232);