Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C221×C4 |
| direct product | C23×#14345 |
| 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 | |
1011144 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)(15,16,18,19)(17,21,22,20)(23,24,27,29)(25,26,30,28);(8,10)(11,12,13,14)(16,19)(17,22)(25,30)(26,28);(3,4);(5,6);(7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,19)(20,22)(23,25)(24,28)(26,29)(27,30);(1,2)(3,4)(5,6);(15,17,18,22)(16,20,19,21)(23,26,27,28)(24,25,29,30)512 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)(15,16,18,19)(17,21,22,20)(23,24,27,29)(25,26,30,28), (8,10)(11,12,13,14)(16,19)(17,22)(25,30)(26,28), (3,4), (5,6), (7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,19)(20,22)(23,25)(24,28)(26,29)(27,30), (1,2)(3,4)(5,6), (15,17,18,22)(16,20,19,21)(23,26,27,28)(24,25,29,30));