Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57710 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,9)(6,13)(7,8)(10,14)(11,17)(12,15)(16,18)(19,20)(21,26)(22,28)(23,25)(24,27)(29,33)(30,34)(31,32)(35,36,40,44)(37,38,42,41)(39,47,49,43)(45,50,48,46);(4,8)(9,15)(10,13)(16,17)(19,21,25,22)(20,26,23,28)(24,30,32,29)(27,34,31,33)(35,37,40,42)(36,41,44,38)(39,48,49,45)(43,46,47,50);(3,5,7,12)(4,9,8,15)(6,11,14,18)(10,16,13,17)(19,22,25,21)(20,26,23,28)(24,30,32,29)(27,33,31,34)(35,37)(36,38)(39,45)(40,42)(41,44)(43,46)(47,50)(48,49);(3,4)(5,9)(6,10)(7,8)(11,16)(12,15)(13,14)(17,18)(19,23,25,20)(21,28,22,26)(24,31,32,27)(29,34,30,33)(35,38)(36,42)(37,44)(39,46)(40,41)(43,48)(45,47)(49,50);(3,6,7,14)(4,10,8,13)(5,11,12,18)(9,16,15,17)(19,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,33)(28,34)(35,39,40,49)(36,43,44,47)(37,45,42,48)(38,46,41,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008933);