Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C225×C4 |
| direct product | C23×#39120 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1589624 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)(13,14)(15,16)(17,18)(20,23)(24,25);(7,8,9,10)(12,16)(14,18)(19,20,22,23)(21,24,26,25);(7,8,9,10)(12,16)(14,18)(19,21)(20,24)(22,26)(23,25);(3,4);(5,6);(7,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,24)(22,26)(23,25);(1,2)(3,4)(5,6);(11,14)(12,17)(13,16)(15,18)(19,20)(21,25)(22,23)(24,26)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((7,8,9,10)(11,12)(13,14)(15,16)(17,18)(20,23)(24,25), (7,8,9,10)(12,16)(14,18)(19,20,22,23)(21,24,26,25), (7,8,9,10)(12,16)(14,18)(19,21)(20,24)(22,26)(23,25), (3,4), (5,6), (7,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,24)(22,26)(23,25), (1,2)(3,4)(5,6), (11,14)(12,17)(13,16)(15,18)(19,20)(21,25)(22,23)(24,26));