#6350973

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

2561 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)(5,11)(6,19)(7,10)(8,14)(13,26)(15,23)(16,25)(17,24)(18,29)(20,22)(21,28)(27,31)(30,32)(33,34)(35,42)(36,41)(37,51)(38,44)(39,45)(40,46)(43,55)(47,60)(48,58)(49,57)(50,64)(52,56)(53,62)(54,61)(59,63);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,28)(18,30)(24,31)(25,32)(26,29)(33,35,40,49)(34,41,46,58)(36,38,48,39)(37,47,55,63)(42,45,57,44)(43,56,51,64)(50,54,52,53)(59,61,60,62);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31)(33,36)(34,42)(35,38)(37,52)(39,49)(40,48)(41,45)(43,60)(44,58)(46,57)(47,53)(50,55)(51,59)(54,63)(56,62)(61,64);(2,10)(4,16)(8,22)(11,19)(12,23)(13,27)(17,18)(28,30)(33,37)(34,43)(35,47)(36,50)(38,54)(39,53)(40,55)(41,56)(42,59)(44,62)(45,61)(46,51)(48,52)(49,63)(57,60)(58,64);(1,5,7,6)(2,11,10,19)(3,14,15,20)(4,17,16,18)(8,23,22,12)(9,24,25,29)(13,28,27,30)(21,31,32,26)(33,38,40,39)(34,44,46,45)(35,48,49,36)(37,53,55,54)(41,42,58,57)(43,61,51,62)(47,50,63,52)(56,60,64,59)

Polycyclic presentation:

2^9|b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258169);