Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#17243 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(8,10)(11,12,15,14)(13,16,17,18)(19,20)(21,25)(22,26)(23,24)(28,29)(30,34);(5,9)(8,10)(12,14)(13,17)(20,24)(22,25)(27,28,31,29)(30,32,34,33);(3,4,6,7)(5,8,9,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,25)(22,24)(23,26)(27,29,31,28)(30,32,34,33);(3,5)(4,8)(6,9)(7,10)(11,14,15,12)(13,16,17,18)(19,22,23,25)(20,26,24,21)(27,30,31,34)(28,32,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752286);