Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#21639 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2446 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);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16,20,18,19,22,28,27)(17,21,25,26,23,29,30,24);(3,4,6,5)(8,11)(12,14)(15,17,19,23)(16,21,22,29)(18,26,27,24)(20,25,28,30);(3,5,6,4)(7,9,10,13)(8,12,11,14)(16,22)(17,23)(18,27)(25,30);(15,18,20,22,19,27,28,16)(17,24,25,21,23,26,30,29)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280642);