Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C22×#99 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C2×Q8 | C24×C19 |
| prime radical | C19×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×Q8×C19 ◁ C2×C19×H(5,𝔽2) ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,31,27,32,23,29)(22,33,34,49,36,44,41,51)(24,30,38,48,25,39,37,47)(28,42,45,40,46,50,43,35);(20,22)(21,28)(23,34)(24,35)(25,40)(26,41)(27,36)(29,43)(30,44)(31,45)(32,46)(33,39)(37,42)(38,50)(47,49)(48,51);(20,23,27,26)(21,29,32,31)(22,34,36,41)(24,38,25,37)(28,43,46,45)(30,48,39,47)(33,49,44,51)(35,50,40,42);(20,24,27,25)(21,30,32,39)(22,35,36,40)(23,37,26,38)(28,44,46,33)(29,47,31,48)(34,42,41,50)(43,49,45,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26359);