#8406492

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

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,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, 8313688);