Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C220 |
| direct product | C2×#7625854 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C26 |
| commutator | C22 | C26×C4 |
| Frattini, cosocle | C23 | C27 |
| prime radical | C22×#3498 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
124049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 18 points:
(8,10)(12,16)(14,17);(7,8)(9,10)(11,12)(13,17)(14,18)(15,16);(3,4,6,5)(12,16)(14,17);(13,18)(14,17);(3,4,6,5)(11,13)(12,17)(14,16)(15,18);(3,5,6,4)(11,14,15,17)(12,13,16,18);(1,2)340 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,10)(12,16)(14,17), (7,8)(9,10)(11,12)(13,17)(14,18)(15,16), (3,4,6,5)(12,16)(14,17), (13,18)(14,17), (3,4,6,5)(11,13)(12,17)(14,16)(15,18), (3,5,6,4)(11,14,15,17)(12,13,16,18), (1,2));