Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C101 · Schur multiplier (H2) = C22×C4 |
| direct product | C101×#9667 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | #121 |
| commutator | C2×C4 | C22×C8×C101 |
| Frattini, cosocle | C2×C42 | C23×C101 |
| socle | C22×C101 | #390 |
| prime radical | C22×D4×C101 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C2×C42×C101 ◁ G | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2,4,7,5,3,6,8)(9,10,14,18,15,17,13,11)(12,20,22,24,23,19,21,16)(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);(1,3,4,8,5,2,6,7)(9,11,14,17,15,18,13,10)(12,19,22,24,23,20,21,16);(9,12,13,21,15,23,14,22)(10,16,18,20,17,24,11,19)6464 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 2655);