#6351930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2559 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259126);