#6349884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2569 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257080);