Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C61 · Schur multiplier (H2) = C25 |
| direct product | C8×#137×C61 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C2×D4 |
| commutator | C4 | C23×C8×C61 |
| Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C2×C4×D4 |
| prime radical | C2×#41×C61 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,3,5,4,6,7,8)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(10,15)(14,23)(16,20)(17,24)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10839);