#6351303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2554 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,27)(15,22)(16,24)(17,25)(18,26)(20,23)(21,29)(28,31)(30,32)(33,34)(35,45)(36,42)(37,40)(38,44)(39,43)(41,48)(46,61)(47,57)(49,54)(50,64)(51,60)(52,55)(53,59)(56,63)(58,62);(1,3,7,15)(2,8,11,23)(4,13,17,30)(5,14,6,20)(9,21,25,31)(10,22,19,12)(16,28,18,29)(24,27,26,32)(34,40)(37,38)(41,54)(42,55)(43,44)(45,57)(47,48)(52,53)(56,64)(58,61)(59,60)(62,63);(1,4)(2,9)(3,13)(5,18)(6,16)(7,17)(8,21)(10,26)(11,25)(12,27)(14,29)(15,30)(19,24)(20,28)(22,32)(23,31)(33,35,39,49)(34,41,43,57)(36,50,51,46)(37,48,38,47)(40,54,44,45)(42,58,60,56)(52,62,53,63)(55,61,59,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,26)(13,28,30,29)(21,27,31,32)(33,36,39,51)(34,42,43,60)(35,46,49,50)(37,52,38,53)(40,55,44,59)(41,56,57,58)(45,61,54,64)(47,62,48,63);(2,11)(8,23)(9,25)(10,19)(12,22)(21,31)(24,26)(27,32)(34,43)(36,51)(40,44)(41,57)(45,54)(46,50)(52,53)(62,63)

Polycyclic presentation:

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

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, 6258499);