#8755066

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:

(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,23)(20,26)(21,25)(22,24)(27,29)(28,32)(30,31)(36,47)(41,50)(44,57)(48,60)(52,55)(54,61)(58,64)(62,63);(4,12)(7,14)(10,16)(13,15)(18,19)(21,29)(24,26)(28,30)(34,39)(36,47)(37,49)(40,43)(44,57)(45,59)(54,61)(62,63);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,18)(20,26,22,24)(21,27,29,25)(28,32,30,31)(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);(17,20,23,22)(18,24,19,26)(21,28,29,30)(25,31,27,32)(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);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,23,29)(18,25,19,27)(20,28,22,30)(24,31,26,32)(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);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,22,23,20)(18,26,19,24)(21,30,29,28)(25,32,27,31)(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=gh,c^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662262);