#499447

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2087
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

549 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:

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

Polycyclic presentation:

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

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, 406643);