Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C4 · Schur multiplier (H2) = C225×C4 |
| direct product | C23×#34194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C22 | C27×C4 |
| Frattini, cosocle | C23 | C28 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1604936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(7,8,10,9)(12,15)(13,17)(20,23)(21,25);(7,8,10,9)(12,15)(13,17)(21,25)(24,26);(19,20)(21,24)(22,23)(25,26);(7,9,10,8)(11,12)(13,18)(14,15)(16,17)(19,20)(21,26)(22,23)(24,25);(3,4);(5,6);(1,2)(3,4)(5,6);(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26)704 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((7,8,10,9)(12,15)(13,17)(20,23)(21,25), (7,8,10,9)(12,15)(13,17)(21,25)(24,26), (19,20)(21,24)(22,23)(25,26), (7,9,10,8)(11,12)(13,18)(14,15)(16,17)(19,20)(21,26)(22,23)(24,25), (3,4), (5,6), (1,2)(3,4)(5,6), (11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26));