Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C24×C19 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,28,34,42,50,43,52,30,32,27,33,29,35,45,54)(26,36,39,49,46,53,41,31,38,48,37,55,40,51,44,47);(20,21,22,23)(24,26)(25,31)(27,41)(28,44)(29,46)(30,38)(32,47)(33,51)(34,53)(35,55)(36,52)(37,43)(39,45)(40,42)(48,54)(49,50);(25,32)(27,28)(31,47)(35,50)(37,39)(41,44)(43,45)(49,55)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5988);