Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7501 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #354 |
| commutator | C42 | C42×C19 |
| Frattini, cosocle | #532 | C22×C19 |
| socle | C2×C19 | #1393 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ C19×#532 ◁ G | |
286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,27,24,28,30,35)(23,26,29,34,32,25,31,33);(21,25,28,26)(22,29,30,31)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 489);