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:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(18,23)(19,21)(25,28)(27,29)(36,47)(41,50)(44,57)(48,60)(52,55)(54,61)(58,64)(62,63);(4,12)(7,14)(10,16)(13,15)(17,18,22,23)(19,24,21,26)(20,28,30,25)(27,32,29,31)(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)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(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);(17,19,22,21)(18,24,23,26)(20,27,30,29)(25,31,28,32)(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)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(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)(17,21,22,19)(18,26,23,24)(20,29,30,27)(25,32,28,31)(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=gh,b^2=g,c^2=hi,d^2=i,e^2=gh,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=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662276);