Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C218×C43 |
| direct product | C23×#14662 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator | C23 | C26×C4 |
| Frattini, cosocle | C24 | C27 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
770440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 7 generators on 30 points:
(8,10)(11,12,13,14)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,29,30,27);(8,10)(11,12,13,14)(16,17)(20,21)(25,30)(27,29);(3,4);(5,6);(7,8,9,10)(11,12)(13,14)(15,17,19,16)(18,20,22,21)(23,24)(25,29)(26,28)(27,30);(1,2)(3,4)(5,6);(15,18)(16,20)(17,21)(19,22)(23,25,26,30)(24,27,28,29)416 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)(11,12,13,14)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,29,30,27), (8,10)(11,12,13,14)(16,17)(20,21)(25,30)(27,29), (3,4), (5,6), (7,8,9,10)(11,12)(13,14)(15,17,19,16)(18,20,22,21)(23,24)(25,29)(26,28)(27,30), (1,2)(3,4)(5,6), (15,18)(16,20)(17,21)(19,22)(23,25,26,30)(24,27,28,29));