Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33041 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16,20,25)(17,26,21,22)(18,23,19,29)(24,28,30,27);(3,4,6,5)(8,11)(12,14)(15,17,20,21)(16,22,25,26)(18,28,19,27)(23,24,29,30);(3,5,6,4)(7,9,10,13)(8,12,11,14)(15,18,20,19)(16,23,25,29)(17,27,21,28)(22,30,26,24);(15,19,20,18)(16,24,25,30)(17,27,21,28)(22,29,26,23)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536620);