Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#99 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C2×Q8 | C24×C19 |
| socle | C2×C19 | C22×#99 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×#41×C19 ◁ G | |
1220 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)(20,21,25,30,26,31,23,28)(22,32,34,49,33,47,40,51)(24,37,36,48,39,29,41,46)(27,42,45,38,43,50,44,35);(20,22)(21,27)(23,34)(24,38)(25,40)(26,33)(28,44)(29,47)(30,45)(31,43)(32,37)(35,39)(36,42)(41,50)(46,51)(48,49);(22,33)(27,43)(32,47)(34,40)(35,38)(42,50)(44,45)(49,51);(20,23,26,25)(21,28,31,30)(22,34,33,40)(24,36,39,41)(27,44,43,45)(29,46,37,48)(32,49,47,51)(35,50,38,42)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26357);