Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11264 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #400 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
578 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)(21,22)(24,25,29,33,30,35,28,34)(26,36,38,31,40,47,39,48)(27,41,43,51,44,50,45,32)(37,49,54,55,42,46,53,52);(20,21,23,22)(24,26,28,39,30,40,29,38)(25,31,34,36,35,48,33,47)(27,42,43,53,44,37,45,54)(32,49,41,55,51,46,50,52);(20,22,23,21)(24,27,28,43,30,44,29,45)(25,32,33,50,35,51,34,41)(26,37,38,53,40,42,39,54)(31,46,36,52,48,49,47,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4252);