#6351834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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
prime radical	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2557 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)(4,18)(5,11)(6,10)(7,12)(8,16)(9,20)(14,22)(15,24)(17,23)(19,26)(21,25)(27,29)(28,31)(30,32)(33,34)(35,44)(36,41)(37,42)(38,52)(39,43)(40,46)(45,60)(47,55)(48,54)(49,56)(50,57)(51,58)(53,61)(59,62)(63,64);(1,3,7,17)(2,8,12,24)(4,14,21,30)(5,15,6,16)(9,22,26,32)(10,23,11,13)(18,31,25,27)(19,28,20,29)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,53)(42,54)(43,55)(44,52)(49,61)(50,63)(51,62)(56,59)(57,60)(58,64);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,29)(16,28)(17,30)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,49)(39,51)(40,53)(42,57)(43,58)(44,59)(46,63)(47,61)(48,62)(52,56)(54,60)(55,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,27,32,31)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,49,51,50)(40,54,55,44)(41,56,58,57)(45,61,62,63)(53,59,64,60);(1,5,7,6)(2,11,12,10)(3,16,17,15)(4,20,21,19)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,43,52)(35,47,48,46)(36,50,51,49)(40,44,55,54)(41,57,58,56)(45,61,62,63)(53,59,64,60)

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6259030);