Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#7274 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,11)(10,13)(12,14)(16,21)(17,25)(18,27)(19,26)(22,23)(24,30)(28,29);(8,11)(9,14)(12,13)(15,16,18,22,19,23,17,21,20,24,25,29,26,28,27,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34899);