Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C101 · Schur multiplier (H2) = C2×C4 |
| direct product | C101×#9724 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | #121 |
| commutator | C42 | C22×C4×C101 |
| Frattini, cosocle | C2×C42 | C23×C101 |
| socle | C22×C101 | #390 |
| prime radical | C101×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C2×C42×C101 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 3 generators on 125 points:
(2,5,6,7)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,2,3,7,4,6,8,5)(10,15,18,17)(16,19,24,23);(9,11,12,20,14,22,13,21)(10,16,17,23,18,24,15,19)5858 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 2712);