Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#21516 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2776 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)(7,8,10,9)(11,12)(13,18)(14,23)(15,20)(16,25)(17,21)(19,24)(22,26);(3,4,5,6)(11,13,17,14)(12,18,21,23)(15,24,16,22)(19,25,26,20);(11,14,17,13)(12,19,21,26)(15,24,16,22)(18,25,23,20);(7,9,10,8)(11,15,17,16)(12,20,21,25)(13,22,14,24)(18,26,23,19)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279998);