Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#732093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
937224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6)(7,8)(9,10)(11,12)(21,22,24,25)(23,26,27,28);(10,12)(14,17)(18,20)(22,25)(23,27);(6,7)(10,12)(15,19)(18,20);(6,7)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(3,4);(1,2)(3,4);(5,7,8,6)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,27)(22,26,25,28)368 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((5,6)(7,8)(9,10)(11,12)(21,22,24,25)(23,26,27,28), (10,12)(14,17)(18,20)(22,25)(23,27), (6,7)(10,12)(15,19)(18,20), (6,7)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28), (3,4), (1,2)(3,4), (5,7,8,6)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,27)(22,26,25,28));