Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#37722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1616616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,25)(23,26);(11,13)(12,15)(14,18)(16,17)(19,21,23,25)(20,22,26,24);(7,8,10,9)(11,12,14,17)(13,15,18,16)(19,21,23,25)(20,24,26,22);(3,4);(5,6);(8,9)(11,12)(13,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25);(1,2)(3,4)(5,6);(7,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,22)(20,25)(21,26)(23,24)464 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((11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,25)(23,26), (11,13)(12,15)(14,18)(16,17)(19,21,23,25)(20,22,26,24), (7,8,10,9)(11,12,14,17)(13,15,18,16)(19,21,23,25)(20,24,26,22), (3,4), (5,6), (8,9)(11,12)(13,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25), (1,2)(3,4)(5,6), (7,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,22)(20,25)(21,26)(23,24));