#6349122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2566 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)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34,39,44)(35,40,47,55)(36,48,50,58)(37,43,38,42)(41,52,56,49)(45,59,61,63)(46,57,51,54)(53,60,64,62);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(33,35)(34,40)(36,45)(37,46)(38,51)(39,47)(41,53)(42,54)(43,57)(44,55)(48,63)(49,60)(50,61)(52,62)(56,64)(58,59);(1,4,6,15)(2,8,9,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(10,23,17,29)(11,25,24,32)(13,27,18,30)(33,36)(34,41)(35,45)(37,49)(38,52)(39,50)(40,53)(42,48)(43,58)(44,56)(46,60)(47,61)(51,62)(54,63)(55,64)(57,59);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,61,62)(53,63,64,59);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)

Polycyclic presentation:

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

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, 6256318);