Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×C19×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
516 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)(21,23)(24,25,27,32,42,51,28,44,30,35,43,52,29,34,41,33)(26,36,37,53,46,48,55,50,40,54,45,49,39,47,38,31);(20,21,22,23)(24,26)(25,31)(27,38)(28,45)(29,46)(30,40)(32,47)(33,36)(34,53)(35,50)(37,41)(39,42)(43,55)(44,54)(48,52)(49,51);(25,32)(27,41)(28,43)(29,42)(31,47)(33,51)(34,44)(35,52)(36,49)(37,38)(39,46)(45,55)(48,50)(53,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6005);