Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13270 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1985 |
| prime radical | C23×C19 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
316 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)(20,21,25,31,26,30,42,45,27,32,43,49,24,38,34,50)(22,33,37,41,36,46,51,39,35,28,44,40,29,47,48,23);(20,22,34,48,24,29,43,44,27,35,42,51,26,36,25,37)(21,28,45,23,38,46,31,40,32,33,50,39,30,47,49,41);(20,23,24,40,27,39,26,41)(21,29,30,22,32,36,38,35)(25,33,34,47,43,28,42,46)(31,48,45,37,49,51,50,44)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6258);