#501404

order	512
nilpotency class	4
isoclinism	isoclinic class of #501404
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24553
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408600);