#6352002

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

2532 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=i,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, 6259198);