Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C23×#59030 |
| 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 | |
1612680 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)(12,15)(13,17)(19,20)(21,25)(22,23)(24,26);(12,15)(16,18);(7,8,9,10)(13,17)(16,18)(20,22)(24,25);(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26);(3,4);(5,6);(7,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22,23,20)(21,25,26,24);(1,2)(3,4)(5,6)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((7,8,9,10)(12,15)(13,17)(19,20)(21,25)(22,23)(24,26), (12,15)(16,18), (7,8,9,10)(13,17)(16,18)(20,22)(24,25), (11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26), (3,4), (5,6), (7,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22,23,20)(21,25,26,24), (1,2)(3,4)(5,6));