#6354905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354835
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1514 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,25)(14,22)(16,26)(17,23)(18,24)(19,30)(21,29)(27,31)(28,32)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,48)(41,52)(46,60)(47,56)(49,55)(51,57)(53,58)(54,61)(59,63)(62,64);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,26)(4,13,19,29,18,28,17,27)(9,21,23,25,24,31,30,32)(33,35,37,48,39,47,38,49)(34,40,43,45,44,55,42,56)(36,46,52,63,53,61,51,62)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,21)(10,23)(11,24)(12,25)(14,27)(15,29)(16,28)(20,30)(22,31)(26,32)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,58)(43,50)(44,57)(45,59)(47,61)(48,63)(49,62)(55,60)(56,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,18,19)(8,12,22,26)(9,23,24,30)(13,27,28,29)(21,25,31,32)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63);(4,18)(9,24)(13,28)(17,19)(21,31)(23,30)(25,32)(27,29)(33,37,39,38)(34,42,44,43)(35,48,47,49)(36,51,53,52)(40,56,55,45)(41,50,57,58)(46,62,61,63)(54,59,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262101);