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×#12634 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#99 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | C4×#99 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C8×C19 ◁ C2×C19×#131 ◁ G | |
452 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)(20,21,25,23,26,29,40,36)(22,30,33,44,34,45,48,51)(24,37,38,49,39,28,43,35)(27,41,31,46,42,50,47,32);(20,22)(21,27)(23,31)(24,32,39,46)(25,33)(26,34)(28,30,37,45)(29,42)(35,44,49,51)(36,47)(38,41,43,50)(40,48);(20,23,25,29,26,36,40,21)(22,31,33,42,34,47,48,27)(24,35,38,37,39,49,43,28)(30,32,44,41,45,46,51,50)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5622);