Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C19 | D4 |
| commutator | C4 | C22×C16×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C4⋊C8 |
| prime radical | C23×C19 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(22,29)(23,27)(24,33)(25,28)(26,31)(30,34)(32,35)(36,37,38,41,39,42,44,47,40,43,45,48,46,49,50,51);(20,22,24,31,25,32,23,30)(21,26,27,29,28,34,33,35);(20,23,25,24)(21,27,28,33)(22,30,32,31)(26,29,34,35)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5000);