Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12963 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | #1867 |
| prime radical | C19×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C4×C19 ◁ C2×#41×C19 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,28,25,29,36,42)(22,30,33,44,31,43,46,51)(23,27,35,41,34,40,47,50)(26,37,39,45,38,48,49,32);(20,22,24,33,25,31,36,46)(21,26,28,39,29,38,42,49)(23,32,35,37,34,45,47,48)(27,30,41,44,40,43,50,51);(22,31)(23,34)(26,38)(30,43)(33,46)(35,47)(39,49)(44,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5951);