Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12762 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C19 |
| Frattini, cosocle | C22×D4 | C23×C19 |
| prime radical | C2×C19×#560 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C19 ◁ C22×D4×C19 ◁ G | |
944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25)(26,32,33,39)(27,35,37,30)(28,31)(29,34)(36,38);(24,26)(25,29)(27,36)(28,33)(30,39)(31,38)(32,35)(34,37);(21,23)(26,33)(29,38)(30,35)(34,36)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5750);