Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C22×#1369 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #106 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | #334 |
| prime radical | C24 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2);(3,4);(5,6,9,15,11,16,26,32,12,17,25,14,10,28,8,24)(7,18,20,29,22,33,19,30,23,31,27,36,21,35,34,13);(5,7,8,19,10,21,25,20,12,23,26,34,11,22,9,27)(6,13,14,18,16,29,24,33,17,30,15,31,28,36,32,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390519);