#8406480

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

3707 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8313676);