Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#617162 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C25×#120 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2761464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6)(7,8)(9,10)(11,12)(14,17)(18,19)(21,22)(23,27)(24,25)(26,28);(6,8)(9,10,11,12)(13,14,16,17)(15,19,20,18)(22,25)(26,27);(9,10)(11,12)(21,23)(22,26)(24,28)(25,27);(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(3,4);(1,2)(3,4);(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28)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)(14,17)(18,19)(21,22)(23,27)(24,25)(26,28), (6,8)(9,10,11,12)(13,14,16,17)(15,19,20,18)(22,25)(26,27), (9,10)(11,12)(21,23)(22,26)(24,28)(25,27), (5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27), (3,4), (1,2)(3,4), (13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28));