#6351210

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
prime radical	#537	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2514 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,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, 6258406);