Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#550 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4702 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)(5,6)(7,8)(9,12)(10,11)(13,14)(16,21)(17,25)(18,20)(22,26)(27,28)(29,30);(8,11)(12,14)(15,16)(17,26)(18,23)(19,21)(20,24)(22,27)(25,29)(28,30);(4,6)(7,9,10,13)(8,12,11,14)(15,17,19,28)(16,22,21,29)(18,27,20,25)(23,30,24,26);(8,11)(12,14)(15,18,19,20)(16,23,21,24)(17,27,28,25)(22,30,29,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347581);