#8926723

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

3703 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)(17,18,23,28)(19,27,22,24)(20,30,29,25)(21,26,31,32)(34,39)(35,43)(37,49)(41,52)(44,46)(47,61)(54,59)(63,64);(1,2,6,4)(3,7,11,9)(5,12,13,8)(10,15,16,14)(17,19,23,22)(18,24,28,27)(20,31,29,21)(25,32,30,26)(34,39)(40,51)(41,52)(42,50)(53,55)(54,59)(56,62)(63,64);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,32)(28,30)(33,34,38,39)(35,40,43,51)(36,41,48,52)(37,50,49,42)(44,53,46,55)(45,59,57,54)(47,62,61,56)(58,64,60,63);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,35,38,43)(34,40,39,51)(36,46,48,44)(37,45,49,57)(41,55,52,53)(42,54,50,59)(47,60,61,58)(56,64,62,63);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,23,31)(18,26,28,32)(19,29,22,20)(24,30,27,25)(33,36,38,48)(34,41,39,52)(35,44,43,46)(37,47,49,61)(40,53,51,55)(42,56,50,62)(45,58,57,60)(54,63,59,64);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,22,23,19)(18,27,28,24)(20,31,29,21)(25,32,30,26)(33,37)(34,42)(35,45)(36,47)(38,49)(39,50)(40,54)(41,56)(43,57)(44,58)(46,60)(48,61)(51,59)(52,62)(53,63)(55,64)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gh,e^2=h,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, 8833919);