#6351875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351869
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259071);