Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12918 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #120 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1850 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
376 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)(20,21,25,31,26,30,42,47,27,32,43,48,24,34,41,49)(22,33,36,39,29,28,44,40,37,45,51,38,35,46,50,23);(20,22,25,36,26,29,42,44,27,37,43,51,24,35,41,50)(21,28,31,40,30,45,47,38,32,46,48,23,34,33,49,39);(20,23)(21,29)(22,34)(24,38)(25,33)(26,39)(27,40)(28,42)(30,37)(31,44)(32,35)(36,49)(41,46)(43,45)(47,51)(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, 5906);