#416799

order	512
nilpotency class	3
isoclinism	isoclinic class of #416472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2864
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15232
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323995);