#9481517

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

3633 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)(3,11)(4,14)(5,8)(6,10)(7,9)(12,15)(13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32)(33,34,39,40)(35,37,42,49)(36,41,48,54)(38,43,53,55)(44,50,57,46)(45,51,52,58)(47,56,62,64)(59,63,61,60);(1,3,8,7)(2,9,5,11)(4,13,6,12)(10,16,14,15)(18,23)(19,26)(25,28)(27,32)(34,40)(37,49)(41,54)(43,55)(46,50)(51,58)(56,64)(60,63);(1,4,8,6)(2,10,5,14)(3,12,7,13)(9,15,11,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35,39,42)(34,37,40,49)(36,44,48,57)(38,52,53,45)(41,50,54,46)(43,58,55,51)(47,61,62,59)(56,60,64,63);(1,5,8,2)(3,11,7,9)(4,10,6,14)(12,15,13,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36)(34,41)(35,44)(37,50)(38,47)(39,48)(40,54)(42,57)(43,56)(45,59)(46,49)(51,63)(52,61)(53,62)(55,64)(58,60);(1,6,8,4)(2,10,5,14)(3,13,7,12)(9,15,11,16)(33,37)(34,42)(35,40)(36,46)(38,51)(39,49)(41,44)(43,52)(45,55)(47,60)(48,50)(53,58)(54,57)(56,59)(61,64)(62,63);(1,7,8,3)(2,11,5,9)(4,13,6,12)(10,16,14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,53)(34,43,40,55)(35,45,42,52)(36,47,48,62)(37,51,49,58)(41,56,54,64)(44,59,57,61)(46,60,50,63)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,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, 9388713);