#416800

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15232
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

619 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:

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

Polycyclic presentation:

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

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, 323996);