Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C4 · Schur multiplier (H2) = C225×C4 |
| direct product | C23×#34493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C22 | C27×C4 |
| Frattini, cosocle | C23 | C28 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1620552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(12,15)(16,18)(19,20)(21,22)(23,24)(25,26);(7,8,10,9)(12,15)(13,17)(19,21)(20,22)(23,26)(24,25);(7,8,10,9)(13,17)(16,18)(20,24)(22,25);(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,24)(23,26);(3,4);(5,6);(1,2)(3,4)(5,6);(7,9,10,8)(11,13,14,17)(12,16,15,18)(19,22,23,25)(20,21,24,26)608 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)(19,20)(21,22)(23,24)(25,26), (7,8,10,9)(12,15)(13,17)(19,21)(20,22)(23,26)(24,25), (7,8,10,9)(13,17)(16,18)(20,24)(22,25), (11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,24)(23,26), (3,4), (5,6), (1,2)(3,4)(5,6), (7,9,10,8)(11,13,14,17)(12,16,15,18)(19,22,23,25)(20,21,24,26));