Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4);(5,6,9,14,11,15,10,26)(7,16,18,33,21,29,20,35)(8,22,23,31,25,36,24,13)(12,27,28,19,30,34,32,17);(5,7,8,19,9,18,23,34,11,21,25,17,10,20,24,27)(6,12,13,29,26,32,36,33,15,30,31,16,14,28,22,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53538);