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×#58470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4956 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)(5,7)(6,9)(8,10)(12,16)(14,17)(19,20,24,27)(21,25,30,34)(22,23,26,33)(28,32,31,29);(4,5)(6,10)(11,12)(13,17)(14,18)(15,16)(19,21,24,30)(20,25,27,34)(22,28,26,31)(23,32,33,29);(4,5)(8,9)(12,16)(13,18)(19,21)(20,25)(22,31)(23,29)(24,30)(26,28)(27,34)(32,33);(3,5,7,4)(6,9,10,8)(11,13)(12,14)(15,18)(16,17)(19,22,24,26)(20,23,27,33)(21,28,30,31)(25,32,34,29);(3,6,7,10)(4,8,5,9)(11,14)(12,13)(15,17)(16,18)(19,23)(20,26)(21,29)(22,27)(24,33)(25,28)(30,32)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011594);