#9481499

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	#54037	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3765 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

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, 9388695);