Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C4⋊C8 |
| prime radical | C23×C19 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,38,41,51)(27,43)(28,34)(29,35)(30,36)(31,37)(32,40,48,46)(33,45)(39,49,53,52)(42,47,54,55)(44,50);(24,26,29,42,30,40,45,53,31,41,44,54,28,46,27,39)(25,32,35,49,36,38,33,47,37,48,50,52,34,51,43,55);(20,22,23,21)(24,27,28,44,31,45,30,29)(25,33,34,35,37,43,36,50)(26,39,46,54,41,53,40,42)(32,47,51,49,48,55,38,52)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5644);