#6357749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264945);