#6499863

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

3742 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)(6,24)(7,12)(8,16)(11,26)(14,39)(15,38)(17,36)(18,31)(19,47)(20,33)(21,50)(22,52)(23,35)(25,29)(27,48)(28,44)(30,41)(32,49)(34,54)(37,42)(40,61)(43,60)(45,59)(46,64)(51,58)(53,56)(55,63)(57,62);(1,3,7,17)(2,8,12,29)(4,14,20,42)(5,15,23,45)(6,25,26,16)(9,30,33,27)(10,28,35,56)(11,36,24,13)(18,40,21,43)(19,48,49,41)(22,53,54,44)(31,57,50,55)(32,37,47,39)(34,59,52,38)(46,63,51,62)(58,60,64,61);(1,4,7,20)(2,9,12,33)(3,14,17,42)(5,21,23,18)(6,19,26,49)(8,27,29,30)(10,31,35,50)(11,32,24,47)(13,37,36,39)(15,43,45,40)(16,41,25,48)(22,51,54,46)(28,55,56,57)(34,58,52,64)(38,60,59,61)(44,63,53,62);(1,5)(2,10)(3,15)(4,18)(6,22)(7,23)(8,28)(9,31)(11,34)(12,35)(13,38)(14,40)(16,44)(17,45)(19,46)(20,21)(24,52)(25,53)(26,54)(27,55)(29,56)(30,57)(32,58)(33,50)(36,59)(37,60)(39,61)(41,62)(42,43)(47,64)(48,63)(49,51);(6,26)(8,29)(11,24)(16,25)(19,49)(22,54)(27,30)(28,56)(32,47)(34,52)(41,48)(44,53)(46,51)(55,57)(58,64)(62,63)

Polycyclic presentation:

2^9|b^2=i,c^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	#72251704, #72263053
maximal normal subgroup of	#72251704, #72263053
maximal quotient of	#72256141, #72256148, #72256161, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407059);