Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2442 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);(7,8)(9,13)(10,11)(12,14)(15,16)(17,19)(18,25)(20,24)(21,27)(22,28)(23,29)(26,30);(3,4,5,6)(8,11)(12,13)(15,17,19,16)(18,21,26,23)(20,24,28,22)(25,29,30,27);(4,6)(15,18,20,27,19,26,28,29)(16,21,22,25,17,23,24,30);(7,9,10,14)(8,12,11,13)(16,17)(18,26)(22,24)(27,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370178);