Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C101 · Schur multiplier (H2) = C23×C4 |
| direct product | C101×#22140 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C101 | C24 |
| commutator | C23 | C23×C4×C101 |
| Frattini, cosocle | C22×C4 | C24×C101 |
| socle | C23×C101 | C23×C4 |
| prime radical | #392×C101 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C101 ◁ G | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,4)(6,9)(7,11)(13,14,18,22,17,21,16,20)(15,19,23,26,24,27,25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(5,6,8,9)(7,10,11,12)(13,14,16,20,17,21,18,22)(15,19,25,28,24,27,23,26);(2,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,27)(22,26);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28)5252 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 15128);