#8926566

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

3730 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833762);