Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #459 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1802 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
400 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)(20,21,26,30,27,32,24,40)(22,31,25,41,35,48,34,28)(23,36,38,51,39,43,37,47)(29,42,44,50,46,49,45,33);(20,22,24,34,27,35,26,25)(21,28,30,31,32,41,40,48)(23,33,37,49,39,50,38,42)(29,43,44,47,46,36,45,51);(20,23)(21,29)(22,33)(24,37)(25,42)(26,38)(27,39)(28,43)(30,44)(31,47)(32,46)(34,49)(35,50)(36,41)(40,45)(48,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5810);