Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C82×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7141 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C19 | D4 |
| commutator | C4 | C82×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C22×C19 | #336 |
| prime radical | C23×C19 | #97 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 2 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(28,29,32,37,33,38,47,55,34,39,48,56,31,36,45,53)(30,40,42,57,41,51,58,54,43,52,46,59,44,35,49,50);(20,21,22,24,23,25,26,27)(28,30)(29,35)(31,44)(32,46)(33,41)(34,43)(36,52)(37,54)(38,40)(39,51)(42,48)(45,58)(47,49)(50,56)(53,57)(55,59)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 129);