#8926718

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

3699 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,e^2=gh,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, 8833914);