#9481508

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

3635 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)(17,18,23,27)(19,20,30,24)(21,25,22,26)(28,31,29,32)(34,39)(35,43)(37,50)(41,52)(45,47)(48,61)(54,58)(63,64);(4,11)(8,14)(10,15)(12,16)(17,19,23,30)(18,24,27,20)(21,28,22,29)(25,32,26,31)(34,39)(40,44)(41,52)(42,51)(53,55)(54,58)(56,62)(63,64);(2,7)(3,9)(8,14)(10,15)(17,19)(18,20)(21,28)(22,29)(23,30)(24,27)(25,31)(26,32)(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)(18,19)(21,31)(22,32)(23,24)(25,28)(26,29)(27,30)(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)(17,21,23,22)(18,25,27,26)(19,28,30,29)(20,31,24,32)(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)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22,23,21)(18,26,27,25)(19,29,30,28)(20,32,24,31)(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=hi,b^2=g,d^2=gi,e^2=gh,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, 9388704);