Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C101 · Schur multiplier (H2) = C23×C4 |
| direct product | C101×#19744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C101 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C101 |
| prime radical | #532×C101 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C101 ◁ C2×C42×C101 ◁ G | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 4 generators on 149 points:
(3,9)(4,10)(6,12)(8,13)(11,15)(14,16)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,44,48)(33,45,42,39)(36,46,47,43)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,44,33,43)(24,36,45,48,38,47,39,40);(1,3,10,15,5,11,4,9)(2,6,13,16,7,14,8,12)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(2,7)(6,14)(8,13)(12,16)(19,30)(20,32)(23,37)(24,38)(27,41)(31,35)(33,42)(39,45)4040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 12732);