Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#11604 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #102 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C2×C19 | C2×D4⋊C8 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
380 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)(20,21,24,30,26,31,41,49,27,32,40,29,25,42,23,38)(22,33,36,28,37,45,39,47,34,50,35,44,43,48,51,46);(20,22)(21,28)(23,39)(24,35)(25,43)(26,37)(27,34)(29,45)(30,48)(31,47)(32,44)(33,49)(36,40)(38,50)(41,51)(42,46);(22,34)(28,44)(33,50)(35,36)(37,43)(39,51)(45,48)(46,47)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4592);