Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#36528 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| prime radical | C23×#2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1585848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(7,8,9,10)(11,12,14,16)(13,17,18,15)(19,20,23,25)(21,24,26,22);(11,13)(12,15)(14,18)(16,17)(19,21,23,26)(20,24,25,22);(7,8,9,10)(12,16)(15,17)(19,22)(20,21)(23,24)(25,26);(3,4);(5,6);(7,8)(9,10)(11,13)(12,17)(14,18)(15,16)(19,22)(20,21)(23,24)(25,26);(1,2)(3,4)(5,6);(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,26)(23,25)560 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((7,8,9,10)(11,12,14,16)(13,17,18,15)(19,20,23,25)(21,24,26,22), (11,13)(12,15)(14,18)(16,17)(19,21,23,26)(20,24,25,22), (7,8,9,10)(12,16)(15,17)(19,22)(20,21)(23,24)(25,26), (3,4), (5,6), (7,8)(9,10)(11,13)(12,17)(14,18)(15,16)(19,22)(20,21)(23,24)(25,26), (1,2)(3,4)(5,6), (11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,26)(23,25));