Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#16595 |
| 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 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,6)(8,9)(11,12)(13,17)(14,16)(15,18)(19,20,22,25)(21,24,26,23);(11,12,14,16)(13,15,18,17)(19,21)(20,23)(22,26)(24,25);(7,8,10,9)(11,12)(13,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(7,9,10,8)(11,13)(12,15)(14,18)(16,17)(19,20)(21,23)(22,25)(24,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518367);