#8416977

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3707 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,32)(29,31)(33,34,38,39)(35,43,46,40)(36,41,49,47)(37,42,51,52)(44,57,59,53)(45,56,50,54)(48,55,62,60)(58,64,61,63);(2,3)(7,9)(8,10)(14,15)(18,23)(24,27)(25,26)(31,32)(34,39)(40,43)(41,47)(42,52)(53,57)(54,56)(55,60)(63,64);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32)(33,35,38,46)(34,40,39,43)(36,44,49,59)(37,50,51,45)(41,53,47,57)(42,56,52,54)(48,61,62,58)(55,64,60,63);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,22,30)(18,24,23,27)(20,28,21,29)(25,31,26,32)(33,36)(34,41)(35,44)(37,48)(38,49)(39,47)(40,53)(42,55)(43,57)(45,58)(46,59)(50,61)(51,62)(52,60)(54,63)(56,64);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,21)(18,25,23,26)(19,28,30,29)(24,31,27,32)(33,34)(35,40)(36,47)(37,42)(38,39)(41,49)(43,46)(44,57)(45,54)(48,60)(50,56)(51,52)(53,59)(55,62)(58,64)(61,63);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,20)(18,26,23,25)(19,29,30,28)(24,32,27,31)(33,37,38,51)(34,42,39,52)(35,45,46,50)(36,48,49,62)(40,54,43,56)(41,55,47,60)(44,58,59,61)(53,63,57,64)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324173);