Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33451 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C22×D8 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,35)(27,33)(28,40)(29,44)(30,32)(31,37)(34,42)(36,52)(38,51)(39,49)(41,47)(43,54)(45,50)(46,53)(48,55);(24,26)(25,31)(27,36)(28,41)(29,45)(30,38)(32,46)(33,47)(34,50)(35,40)(37,42)(39,53)(43,55)(44,48)(49,52)(51,54);(20,22,23,21)(25,32)(31,46)(33,44)(34,49)(35,51)(40,54)(47,48)(50,52);(24,27,30,29)(25,33,32,44)(26,36,38,45)(28,39,43,42)(31,47,46,48)(34,40,49,54)(35,52,51,50)(37,41,53,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26439);