#8926564

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833760);