#6352929

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2603 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260125);