Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#23833 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | #2398 |
| prime radical | C19×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 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)(22,31,34,44)(23,28)(24,29)(25,39)(26,30)(27,35,41,32)(33,45,47,51)(36,42)(37,49)(38,43)(40,48,50,46);(22,32,34,35)(27,31,41,44)(33,46,47,48)(40,45,50,51);(20,22)(21,27)(23,33)(24,32)(25,35)(26,34)(28,40)(29,44)(30,41)(31,39)(36,46)(37,48)(38,47)(42,51)(43,50)(45,49);(20,23,24,36,26,38,25,37)(21,28,29,42,30,43,39,49)(22,33,32,46,34,47,35,48)(27,40,44,51,41,50,31,45)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16821);