Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10694 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(4,6)(7,8,9,10)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(11,12,13,14)(17,23)(20,26)(24,29)(27,30)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260306);