Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#17272 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4546 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)(8,10)(11,12,14,15)(13,16,18,17)(19,20)(21,22)(23,24)(25,26)(28,31)(32,33);(5,9)(8,10)(12,15)(16,17)(20,24)(21,26)(27,28,30,31)(29,32,34,33);(3,4,6,7)(5,8,9,10)(11,13,14,18)(12,16,15,17)(19,21)(20,25)(22,24)(23,26)(27,28)(29,33)(30,31)(32,34);(3,5,6,9)(4,8,7,10)(11,12)(13,17)(14,15)(16,18)(19,22,23,25)(20,26,24,21)(27,29,30,34)(28,32,31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752319);