Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#45444 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1723872 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)(19,20,22,24)(21,25,26,23);(11,13)(12,15)(14,18)(16,17)(19,20)(21,25)(22,24)(23,26);(7,8,9,10)(11,13,14,18)(12,15,16,17)(19,20,22,24)(21,23,26,25);(11,12)(13,15)(14,16)(17,18)(19,21)(20,23)(22,26)(24,25);(3,4);(5,6);(7,8)(9,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,23)(22,24)(25,26);(1,2)(3,4)(5,6)416 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)(19,20,22,24)(21,25,26,23), (11,13)(12,15)(14,18)(16,17)(19,20)(21,25)(22,24)(23,26), (7,8,9,10)(11,13,14,18)(12,15,16,17)(19,20,22,24)(21,23,26,25), (11,12)(13,15)(14,16)(17,18)(19,21)(20,23)(22,26)(24,25), (3,4), (5,6), (7,8)(9,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,23)(22,24)(25,26), (1,2)(3,4)(5,6));