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×#58590 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,6,9)(5,8,10,7)(11,12)(13,17)(14,18)(15,16);(3,5,6,10)(4,7,9,8)(11,13,15,18)(12,14,16,17)(20,24)(22,30)(25,27)(28,34);(5,10)(7,8)(12,16)(14,17)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(3,5)(4,8)(6,10)(7,9)(11,14)(12,13)(15,17)(16,18)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,4)(5,8)(6,9)(7,10)(11,13)(12,14)(15,18)(16,17)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011970);