#6469079

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34963
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 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)(3,17)(4,23)(5,12)(6,13)(7,15)(8,14)(9,16)(10,21)(11,26)(18,54)(19,52)(20,53)(22,49)(24,37)(25,47)(27,50)(28,61)(29,43)(30,42)(31,45)(32,62)(33,48)(34,41)(35,40)(36,44)(38,59)(39,60)(46,55)(51,57)(56,63)(58,64);(1,3)(2,10)(4,24)(5,28)(6,32)(7,35)(8,21)(9,22)(11,37)(12,38)(13,46)(14,49)(15,17)(16,40)(18,27)(19,31)(20,34)(23,51)(25,55)(26,56)(29,57)(30,58)(33,60)(36,59)(39,41)(42,61)(43,54)(44,52)(45,64)(47,53)(48,62)(50,63);(1,4,8,26,9,27,7,29)(2,11,15,23,16,43,14,50)(3,18,21,57,22,24,35,56)(5,25,30,34,31,33,36,6)(10,37,17,51,40,54,49,63)(12,41,44,47,45,13,42,48)(19,55,59,20,28,60,58,32)(38,39,52,53,64,46,61,62);(1,5)(2,12)(3,19)(4,25)(6,29)(7,36)(8,30)(9,31)(10,38)(11,41)(13,43)(14,42)(15,44)(16,45)(17,52)(18,55)(20,57)(21,59)(22,28)(23,47)(24,60)(26,34)(27,33)(32,56)(35,58)(37,39)(40,64)(46,54)(48,50)(49,61)(51,53)(62,63);(1,6,7,33,9,34,8,25)(2,13,15,48,16,41,14,47)(3,20,35,55,22,32,21,60)(4,5,29,36,27,31,26,30)(10,39,17,53,40,46,49,62)(11,42,23,12,43,44,50,45)(18,28,56,59,24,19,57,58)(37,52,51,64,54,61,63,38)

Polycyclic presentation:

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

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, 6376275);