Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#1426911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C2×#14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
38637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(8,10)(11,12)(13,17)(14,18)(15,16);(4,6)(7,8)(9,10)(11,13,15,18)(12,14,16,17)(19,20,22,24)(21,23,26,25);(4,6)(12,16)(14,17)(19,20,22,24)(21,25,26,23);(1,2);(3,4,5,6)(11,14)(12,13)(15,17)(16,18)(19,20)(21,23)(22,24)(25,26);(11,13)(12,17)(14,16)(15,18)(19,21)(20,23)(22,26)(24,25)184 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,10)(11,12)(13,17)(14,18)(15,16), (4,6)(7,8)(9,10)(11,13,15,18)(12,14,16,17)(19,20,22,24)(21,23,26,25), (4,6)(12,16)(14,17)(19,20,22,24)(21,25,26,23), (1,2), (3,4,5,6)(11,14)(12,13)(15,17)(16,18)(19,20)(21,23)(22,24)(25,26), (11,13)(12,17)(14,16)(15,18)(19,21)(20,23)(22,26)(24,25));