Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#617804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
41947 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)(11,12,15,13)(14,16,18,17)(19,20,22,25)(21,23,26,24);(7,8)(9,10)(11,12,15,13)(14,17,18,16)(19,21,22,26)(20,23,25,24);(19,21)(20,23)(22,26)(24,25);(1,2);(4,6)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,25)(24,26)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)(11,12,15,13)(14,16,18,17)(19,20,22,25)(21,23,26,24), (7,8)(9,10)(11,12,15,13)(14,17,18,16)(19,21,22,26)(20,23,25,24), (19,21)(20,23)(22,26)(24,25), (1,2), (4,6)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,26)(23,25), (3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,25)(24,26));