Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12944 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1862 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
388 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)(20,21,25,31,27,32,26,45)(22,33,37,49,38,29,42,48)(23,39,36,47,40,46,35,28)(24,41,34,50,44,51,43,30);(20,22,25,37,27,38,26,42)(21,28,31,39,32,47,45,46)(23,34,36,44,40,43,35,24)(29,30,48,41,33,50,49,51);(20,23)(21,29)(22,34)(24,42)(25,36)(26,35)(27,40)(28,30)(31,48)(32,33)(37,44)(38,43)(39,41)(45,49)(46,51)(47,50)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5932);