Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C101 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C101×#1938 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C101 | C2×D4 |
| commutator | C2×C4 | C23×C4×C101 |
| Frattini, cosocle | C22×C4 | C24×C101 |
| socle | C23×C101 | C4×D4 |
| prime radical | C22×#41×C101 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C101 ◁ C23×C8×C101 ◁ G | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(3,4)(5,6)(7,8,11,15,12,16,10,13)(9,17,20,14,18,21,19,22);(8,13,16,15)(9,18)(14,21,22,17)(19,20);(4,6)(7,9,10,19,12,18,11,20)(8,14,15,21,16,22,13,17)6464 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 14085);