#6351858

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	#60961	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2558 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)(38,51)(39,43)(40,46)(45,60)(47,54)(48,56)(49,58)(50,57)(52,55)(53,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28)(20,29)(21,30)(25,27)(26,32)(33,35)(34,40)(36,45)(37,46)(38,52)(39,47)(41,53)(42,54)(43,55)(44,51)(48,61)(49,63)(50,62)(56,59)(57,64)(58,60);(1,4,7,21)(2,9,12,26)(3,14,17,30)(5,20,6,19)(8,22,24,32)(10,18,11,25)(13,27,23,31)(15,29,16,28)(33,36,39,50)(34,41,43,57)(35,45,47,62)(37,49,38,48)(40,53,55,64)(42,58,51,56)(44,59,54,60)(46,63,52,61);(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)(34,42,43,51)(35,46,47,52)(36,48,50,49)(40,54,55,44)(41,56,57,58)(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)(35,47)(40,55)(44,54)(45,62)(46,52)(53,64)(59,60)(61,63)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=hi,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, 6259054);