#6366764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#18742	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273960);