#8417308

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

3713 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)(4,10)(8,14)(12,16)(17,18)(19,20)(21,26)(22,27)(23,24)(25,32)(28,30)(29,31)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(3,11)(8,14)(12,16)(13,15)(18,24)(19,25)(26,27)(28,29)(33,34,38,42)(35,44,43,39)(36,47,50,40)(37,41,52,51)(45,53,48,55)(46,58,57,54)(49,60,62,56)(59,63,61,64);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,15)(13,16)(17,19,23,25)(18,20,24,32)(21,28,22,29)(26,30,27,31)(33,34)(35,39)(36,40)(37,51)(38,42)(41,52)(43,44)(45,53)(46,58)(47,50)(48,55)(49,60)(54,57)(56,62)(59,64)(61,63);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,15)(10,16)(17,20)(18,25)(19,24)(21,30)(22,31)(23,32)(26,29)(27,28)(33,35,38,43)(34,39,42,44)(36,48,50,45)(37,46,52,57)(40,55,47,53)(41,54,51,58)(49,61,62,59)(56,64,60,63);(1,4,6,10)(2,9,7,5)(3,12,11,16)(8,15,14,13)(17,21,23,22)(18,26,24,27)(19,28,25,29)(20,30,32,31)(33,36,38,50)(34,40,42,47)(35,45,43,48)(37,49,52,62)(39,53,44,55)(41,56,51,60)(46,59,57,61)(54,63,58,64);(1,5,6,9)(2,10,7,4)(3,13,11,15)(8,16,14,12)(17,22,23,21)(18,27,24,26)(19,29,25,28)(20,31,32,30)(33,37)(34,41)(35,46)(36,49)(38,52)(39,54)(40,56)(42,51)(43,57)(44,58)(45,59)(47,60)(48,61)(50,62)(53,63)(55,64)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gi,e^2=hi,f^2=gh,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, 8324504);