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×#16329 |
| 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 | |
3550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,6)(11,12)(13,17)(14,18)(15,16)(19,20,23,26)(21,22,25,24);(8,10)(12,16)(13,18)(19,21,23,25)(20,24,26,22);(7,8)(9,10)(11,13,15,18)(12,17,16,14)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,17)(12,13,16,18)(19,22,23,24)(20,25,26,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518008);