#6357756

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
prime radical	#3511	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1728 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)(4,16)(5,9)(6,20)(7,11)(8,14)(10,21)(13,27)(15,28)(17,26)(18,25)(19,31)(22,32)(23,29)(24,30)(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);(1,3,6,14,7,15,21,30)(2,8,10,12,11,24,20,28)(4,17,18,32,5,13,19,29)(9,23,25,27,16,22,31,26)(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);(1,4,7,5)(2,9,11,16)(3,13,15,17)(6,18,21,19)(8,22,24,23)(10,25,20,31)(12,26,28,27)(14,29,30,32)(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);(1,5,7,4)(2,9,11,16)(3,13,15,17)(6,19,21,18)(8,23,24,22)(10,25,20,31)(12,27,28,26)(14,29,30,32)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(4,5)(9,16)(13,17)(18,19)(22,23)(25,31)(26,27)(29,32)(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|b^2=f,c^2=g,d^2=h,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, 6264952);