#6465518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2537
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2447 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)(3,14)(4,20)(5,10)(6,12)(7,13)(8,17)(9,24)(11,28)(15,29)(16,42)(18,32)(19,31)(21,33)(22,49)(23,36)(25,30)(26,39)(27,40)(34,53)(35,54)(37,47)(38,55)(41,45)(43,56)(44,48)(46,57)(50,58)(51,59)(52,61)(60,64)(62,63);(1,3)(2,8)(4,21)(5,25)(6,18)(7,19)(9,33)(10,37)(11,31)(12,14)(13,32)(15,24)(16,26)(17,28)(20,48)(22,51)(23,45)(27,46)(29,36)(30,39)(34,59)(35,41)(38,57)(40,42)(43,52)(44,54)(47,55)(49,62)(50,53)(56,60)(58,61)(63,64);(1,4,7,23)(2,9)(3,15)(5,22,27,52)(6,24,28,54)(8,29,32,48)(10,34)(11,20)(12,36)(13,35)(14,41,31,33)(16,43)(17,21)(18,45)(19,44)(25,50)(26,53,55,64)(30,56,57,59)(37,58,42,62)(38,49)(39,61)(40,60)(46,63)(47,51);(1,5,6,26,7,27,28,55)(2,10,12,39,13,40,11,38)(3,16,17,25,19,47,18,46)(4,22,24,53,23,52,54,64)(8,30,31,37,32,57,14,42)(9,34,36,61,35,60,20,49)(15,43,21,50,44,51,45,63)(29,56,33,58,48,59,41,62);(4,23)(9,35)(15,44)(20,36)(21,45)(22,52)(24,54)(29,48)(33,41)(34,60)(43,51)(49,61)(50,63)(53,64)(56,59)(58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372714);