Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#22056 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(7,8)(9,13)(10,12)(11,14)(15,16,19,23)(17,20,24,28)(18,21,26,22)(25,27,30,29);(3,5,6,4)(7,9)(8,11)(10,14)(12,13)(15,17)(16,20)(18,25)(19,24)(21,27)(22,29)(23,28)(26,30);(9,14)(11,13)(16,21)(18,26)(20,27)(22,23)(25,30)(28,29);(8,12)(11,13)(17,24)(20,28)(25,30)(27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282696);