Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C221 |
| direct product | C23×#3003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C27 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
478765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 18 points:
(1,2);(3,4);(5,6);(12,14)(13,16)(17,18);(11,12)(13,17)(14,15);(7,8)(9,10);(8,10)(11,13)(12,16)(14,18)(15,17)256 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), (5,6), (12,14)(13,16)(17,18), (11,12)(13,17)(14,15), (7,8)(9,10), (8,10)(11,13)(12,16)(14,18)(15,17));