Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#59420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1608584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(12,15)(16,18)(19,20)(21,23)(22,24)(25,26);(8,10)(12,15)(16,18)(19,21)(20,23)(22,26)(24,25);(8,10)(20,24)(23,25);(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,26)(23,24);(3,4);(5,6);(1,2)(3,4)(5,6);(7,8,9,10)(11,13,14,17)(12,16,15,18)(19,20)(21,25)(22,24)(23,26)512 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((12,15)(16,18)(19,20)(21,23)(22,24)(25,26), (8,10)(12,15)(16,18)(19,21)(20,23)(22,26)(24,25), (8,10)(20,24)(23,25), (11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,26)(23,24), (3,4), (5,6), (1,2)(3,4)(5,6), (7,8,9,10)(11,13,14,17)(12,16,15,18)(19,20)(21,25)(22,24)(23,26));