Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C2×#7659580 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C26 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
48533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2)(4,7)(8,10)(11,12,16,13)(14,17,18,15)(19,20)(21,24)(22,26)(23,25);(4,7)(8,10)(19,21)(20,24)(22,25)(23,26);(4,7)(8,10)(21,26)(22,24);(4,7)(8,10)(11,13,16,12)(14,15,18,17)(19,20)(21,24)(22,26)(23,25);(5,9)(8,10)(19,22)(20,21)(23,24)(25,26);(3,4)(5,10)(6,7)(8,9)(11,14)(12,17)(13,15)(16,18);(3,5)(4,8)(6,9)(7,10)(11,15)(12,14)(13,18)(16,17)(19,20)(21,24)(22,26)(23,25)196 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((1,2)(4,7)(8,10)(11,12,16,13)(14,17,18,15)(19,20)(21,24)(22,26)(23,25), (4,7)(8,10)(19,21)(20,24)(22,25)(23,26), (4,7)(8,10)(21,26)(22,24), (4,7)(8,10)(11,13,16,12)(14,15,18,17)(19,20)(21,24)(22,26)(23,25), (5,9)(8,10)(19,22)(20,21)(23,24)(25,26), (3,4)(5,10)(6,7)(8,9)(11,14)(12,17)(13,15)(16,18), (3,5)(4,8)(6,9)(7,10)(11,15)(12,14)(13,18)(16,17)(19,20)(21,24)(22,26)(23,25));