#8926562

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3648 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)(17,18,24,28)(19,29,31,22)(20,30,32,25)(21,27,23,26)(33,34,39,40)(35,37,42,49)(36,41,48,54)(38,43,53,55)(44,50,57,46)(45,51,52,58)(47,56,62,64)(59,63,61,60);(5,13)(8,12)(10,16)(14,15)(17,19,24,31)(18,22,28,29)(20,27,32,26)(21,25,23,30)(34,40)(37,49)(41,54)(43,55)(46,50)(51,58)(56,64)(60,63);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,24,32)(18,25,28,30)(19,26,31,27)(21,29,23,22)(33,35,39,42)(34,37,40,49)(36,44,48,57)(38,52,53,45)(41,50,54,46)(43,58,55,51)(47,61,62,59)(56,60,64,63);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,24,23)(18,26,28,27)(19,25,31,30)(20,29,32,22)(33,36)(34,41)(35,44)(37,50)(38,47)(39,48)(40,54)(42,57)(43,56)(45,59)(46,49)(51,63)(52,61)(53,62)(55,64)(58,60);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,19)(20,21)(23,32)(24,29)(25,26)(27,30)(28,31)(33,37)(34,42)(35,40)(36,46)(38,51)(39,49)(41,44)(43,52)(45,55)(47,60)(48,50)(53,58)(54,57)(56,59)(61,64)(62,63);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,23,24,21)(18,27,28,26)(19,30,31,25)(20,22,32,29)(33,38,39,53)(34,43,40,55)(35,45,42,52)(36,47,48,62)(37,51,49,58)(41,56,54,64)(44,59,57,61)(46,60,50,63)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,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, 8833758);