#6499864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#35346
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dhi,g^d=gi,f^e=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263053
maximal normal subgroup of	#72263053
maximal quotient of	#72263055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407060);