Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#12002 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #106 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C2×C19 | C2×#334 |
| prime radical | C23×C19 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
220 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)(20,21,24,30,26,31,41,47,27,32,40,29,25,43,23,39)(22,33,36,45,38,48,35,44,34,46,42,51,37,50,49,28);(20,22,23,35,25,37,40,36,27,34,41,49,26,38,24,42)(21,28,29,33,31,45,39,48,32,44,30,46,43,51,47,50);(22,34)(28,44)(33,46)(35,49)(36,42)(37,38)(45,51)(48,50)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4990);