#8406469

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

3711 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:

(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(18,23)(20,26)(24,30)(28,32)(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);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(19,27)(24,30)(28,32)(29,31)(35,43)(36,46)(37,49)(40,54)(45,52)(48,57)(51,61)(56,64);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32)(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);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,27)(18,24,23,30)(20,28,26,32)(21,29,25,31)(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);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,31,30,29)(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);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)(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=i,b^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, 8313665);