Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12720 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #410 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | #1782 |
| prime radical | C19×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C42×C19 ◁ C4×#41×C19 ◁ G | |
694 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)(22,31)(23,36,38,28)(24,29)(25,30)(26,33,42,32)(27,37,44,51)(34,43)(35,47)(39,45)(40,50)(41,49,46,48);(20,22)(21,26)(23,33)(24,34)(25,35)(27,43)(28,31)(29,41)(30,42)(32,38)(36,47)(37,49)(39,50)(40,44)(45,46)(48,51);(20,23)(21,27,30,44)(22,32)(24,37)(25,38)(26,40,42,43)(28,29,36,45)(31,46,47,41)(33,35)(34,48)(39,51)(49,50)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5708);