#8932120

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

3701 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)(3,9,11,4)(5,13,14,8)(10,12,16,15)(19,26)(22,24)(27,32)(29,31)(34,39)(35,43)(37,50)(41,52)(45,47)(48,61)(54,58)(63,64);(2,7)(4,9)(8,13)(12,15)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(34,39)(40,44)(41,52)(42,51)(53,55)(54,58)(56,62)(63,64);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(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);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(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)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(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);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(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=hi,c^2=gh,d^2=gh,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, 8839316);