Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#59006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| prime radical | C25×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3451664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 26 points:
(12,15)(16,18)(20,23)(24,26);(7,8,9,10)(12,15)(13,17)(21,25)(24,26);(19,20)(21,24)(22,23)(25,26);(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26);(3,4);(5,6);(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26);(1,2)(3,4)(5,6)512 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((12,15)(16,18)(20,23)(24,26), (7,8,9,10)(12,15)(13,17)(21,25)(24,26), (19,20)(21,24)(22,23)(25,26), (11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26), (3,4), (5,6), (7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26), (1,2)(3,4)(5,6));