Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#2099247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
40299 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)(11,12,15,14)(13,17,18,16)(19,20)(21,25)(22,23)(24,26);(4,6)(7,8)(9,10)(11,13,15,18)(12,16,14,17)(19,20)(21,24)(22,23)(25,26);(4,6)(12,14)(16,17)(19,21,23,26)(20,24,22,25);(1,2);(3,4)(5,6)(11,14,15,12)(13,16,18,17)(19,22,23,20)(21,25,26,24);(11,13)(12,16)(14,17)(15,18)(19,21)(20,25)(22,24)(23,26)160 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)(11,12,15,14)(13,17,18,16)(19,20)(21,25)(22,23)(24,26), (4,6)(7,8)(9,10)(11,13,15,18)(12,16,14,17)(19,20)(21,24)(22,23)(25,26), (4,6)(12,14)(16,17)(19,21,23,26)(20,24,22,25), (1,2), (3,4)(5,6)(11,14,15,12)(13,16,18,17)(19,22,23,20)(21,25,26,24), (11,13)(12,16)(14,17)(15,18)(19,21)(20,25)(22,24)(23,26));