Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13129 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1924 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
306 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)(20,21,25,30,26,31,42,33,24,29,32,36,39,45,47,49)(22,23,28,37,34,38,44,48,35,27,40,41,46,43,50,51);(20,22,32,28,26,34,47,44,24,35,25,40,39,46,42,50)(21,27,30,37,31,43,33,48,29,23,36,41,45,38,49,51);(20,23,24,27)(21,28)(22,33)(25,41,32,37)(26,38,39,43)(29,40)(30,46)(31,44)(34,36)(35,49)(42,51,47,48)(45,50)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6117);