#499400

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=eh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fh,g^c=gi,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, 406596);