#6355420

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355407
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
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

1487 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)(3,12,16,26)(4,9,17,23)(5,10,6,20)(8,14,22,15)(13,25,27,32)(18,24,19,30)(21,28,31,29)(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);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,26)(4,13,19,29,17,27,18,28)(9,21,24,25,23,31,30,32)(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);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,28)(15,29)(16,27)(20,30)(22,31)(26,32)(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);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,26)(9,24,23,30)(13,28,27,29)(21,25,31,32)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,d^c=dh,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, 6262616);