#8932118

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3631 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:

(3,10)(6,8)(11,16)(13,15)(17,18,24,28)(19,29,31,22)(20,30,32,25)(21,27,23,26)(34,39)(35,43)(37,50)(41,52)(45,47)(48,61)(54,58)(63,64);(1,2,7,4)(3,8,10,6)(5,12,14,9)(11,13,16,15)(17,19,24,31)(18,22,28,29)(20,27,32,26)(21,25,23,30)(34,39)(40,44)(41,52)(42,51)(53,55)(54,58)(56,62)(63,64);(1,3,7,10)(2,8,4,6)(5,11,14,16)(9,15,12,13)(17,20,24,32)(18,25,28,30)(19,26,31,27)(21,29,23,22)(33,34,38,39)(35,44,43,40)(36,41,49,52)(37,51,50,42)(45,55,47,53)(46,54,57,58)(48,62,61,56)(59,63,60,64);(1,4,7,2)(3,6,10,8)(5,12,14,9)(11,13,16,15)(17,21,24,23)(18,26,28,27)(19,25,31,30)(20,29,32,22)(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);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,19)(20,21)(23,32)(24,29)(25,26)(27,30)(28,31)(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);(1,6)(2,3)(4,10)(5,13)(7,8)(9,11)(12,16)(14,15)(17,23,24,21)(18,27,28,26)(19,30,31,25)(20,22,32,29)(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=g,b^2=ghi,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839314);