Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33463 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | C2×D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×C4×D8 |
| prime radical | C19×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
652 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)(20,21,25,30,26,29,41,31)(22,32,34,43,36,49,35,51)(23,28,37,47,38,46,24,39)(27,42,45,50,44,40,48,33);(20,22)(21,27)(23,33)(24,40)(25,34)(26,36)(28,43)(29,44)(30,45)(31,48)(32,39)(35,41)(37,42)(38,50)(46,51)(47,49);(20,23,25,37,26,38,41,24)(21,28,30,47,29,46,31,39)(22,33,34,42,36,50,35,40)(27,43,45,49,44,51,48,32);(21,29)(27,44)(28,46)(30,31)(32,49)(39,47)(43,51)(45,48)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26451);