Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#16719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(5,9)(11,12,15,13)(14,16,18,17)(19,20,22,24)(21,25,26,23)(28,31)(32,33);(4,7)(8,10)(12,13)(16,17)(27,28,30,31)(29,32,34,33);(3,4,6,7)(5,8,9,10)(11,13,15,12)(14,16,18,17)(19,21,22,26)(20,23,24,25)(27,28)(29,33)(30,31)(32,34);(3,5)(4,8)(6,9)(7,10)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,32)(30,34)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518647);