Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C2×C19×#1321 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #99 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C2×M16 | C23×C19 |
| socle | C22×C19 | #323 |
| prime radical | C22×#41×C19 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(3,4,7,12,8,11,22,28,9,13,23,30,6,18,20,29)(5,14,16,31,15,10,24,25,17,27,21,33,19,26,32,34);(3,5)(4,10)(6,19)(7,21)(8,15)(9,17)(11,27)(12,25)(13,26)(14,18)(16,23)(20,24)(22,32)(28,33)(29,31)(30,34)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4425);