Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| 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.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,8,7)(3,12,6,9)(4,11,5,10)(13,15,14,16)(17,18,23,24)(19,21,26,22)(20,28,31,25)(27,30,32,29)(37,49)(42,51)(45,59)(48,60)(53,56)(54,61)(58,64)(62,63);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(18,24)(19,26)(25,28)(27,32)(34,39)(36,47)(40,43)(42,51)(44,57)(48,60)(53,56)(58,64);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19)(18,21)(20,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,55)(45,56)(47,50)(48,54)(49,51)(52,57)(53,59)(58,63)(60,61)(62,64);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,20,23,31)(18,25,24,28)(19,27,26,32)(21,29,22,30)(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);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,21,23,22)(18,26,24,19)(20,29,31,30)(25,32,28,27)(33,36,38,47)(34,41,39,50)(35,44,46,57)(37,48,49,60)(40,52,43,55)(42,54,51,61)(45,58,59,64)(53,62,56,63);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,22,23,21)(18,19,24,26)(20,30,31,29)(25,27,28,32)(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=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021581);