#8093502

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000698);