Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C101 · Schur multiplier (H2) = C23 |
| direct product | C2×C101×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C101 | C22⋊C4 |
| commutator | C22 | C22×C16×C101 |
| Frattini, cosocle | C2×C16 | C23×C101 |
| socle | C22×C101 | C22⋊C16 |
| prime radical | C24×C101 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C101 ◁ C22×C16×C101 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(3,4,5,10,6,11,14,22,7,12,15,23,16,24,27,32,8,9,13,19,17,20,25,29,18,21,26,30,28,31,33,34);(4,9)(5,13)(11,20)(12,21)(14,25)(15,26)(24,31)(27,33)8888 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 6543);