Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C101 · Schur multiplier (H2) = C2 |
| direct product | C101×#12053 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C101 | C2×D4 |
| commutator | C2×C4 | C22×C8×C101 |
| Frattini, cosocle | C4×C8 | C23×C101 |
| socle, prime radical | C22×C101 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C101 ◁ C4×C16×C101 ◁ G | |
206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 25856 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,20,24,21,19,26,27,22,25,29,32,30,28,23,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9);(17,19,20,27,21,25,26,32,22,28,29,31,30,18,23,24)7676 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 5041);