Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25225 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,23)(19,22)(20,24)(25,30)(26,28)(27,29);(3,5,6,4)(8,11)(12,14)(15,17,19,27)(16,21,22,29)(18,26,20,25)(23,28,24,30);(4,5)(7,9,10,13)(8,12,11,14)(16,22)(17,25)(18,20)(21,28)(26,27)(29,30);(8,11)(12,14)(15,18,19,20)(16,23,22,24)(17,26,27,25)(21,28,29,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348808);