Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12646 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #102 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | C4×#102 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,31,27,33,26,43)(22,34,37,46,40,49,39,44)(23,29,41,48,42,32,25,30)(28,38,45,35,47,51,50,36);(20,22,25,38,26,39,42,36,27,40,41,51,24,37,23,35)(21,28,32,46,31,45,30,49,33,47,29,44,43,50,48,34);(20,23,24,41,27,42,26,25)(21,29,31,48,33,32,43,30)(22,35,37,51,40,36,39,38)(28,44,45,34,47,46,50,49)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5634);