Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C2×#7837040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C26 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
50411 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)(3,4)(5,8)(6,10)(7,9)(12,15)(13,17)(20,22)(21,26);(3,5,7,10)(4,8,9,6)(11,12,14,15)(13,16,17,18)(19,20,23,22)(21,25,26,24);(3,5,7,10)(4,8,9,6)(11,12,14,15)(13,18,17,16)(19,21,23,26)(20,24,22,25);(11,12,14,15)(13,16,17,18)(19,21,23,26)(20,25,22,24);(13,17)(16,18)(19,21)(20,24)(22,25)(23,26);(3,4)(5,8)(6,10)(7,9)(11,12)(13,18)(14,15)(16,17)(19,21)(20,24)(22,25)(23,26);(3,6,7,8)(4,5,9,10)(11,13,14,17)(12,16,15,18)(19,22,23,20)(21,24,26,25)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)(3,4)(5,8)(6,10)(7,9)(12,15)(13,17)(20,22)(21,26), (3,5,7,10)(4,8,9,6)(11,12,14,15)(13,16,17,18)(19,20,23,22)(21,25,26,24), (3,5,7,10)(4,8,9,6)(11,12,14,15)(13,18,17,16)(19,21,23,26)(20,24,22,25), (11,12,14,15)(13,16,17,18)(19,21,23,26)(20,25,22,24), (13,17)(16,18)(19,21)(20,24)(22,25)(23,26), (3,4)(5,8)(6,10)(7,9)(11,12)(13,18)(14,15)(16,17)(19,21)(20,24)(22,25)(23,26), (3,6,7,8)(4,5,9,10)(11,13,14,17)(12,16,15,18)(19,22,23,20)(21,24,26,25));