Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219×C42 |
| direct product | C22×#681049 |
| 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 | |
956808 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)(15,19)(17,20)(24,28)(26,27);(5,6,7,8)(9,10)(11,12)(15,19)(17,20)(21,22)(23,25)(24,27)(26,28);(6,8)(10,12)(22,23)(24,28);(5,6)(7,8)(9,10,11,12)(13,14,16,18)(15,17,19,20)(21,23,25,22)(24,27,28,26);(3,4);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12)(13,15,16,19)(14,17,18,20)(21,24,25,28)(22,26,23,27)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((5,6)(7,8)(9,10,11,12)(15,19)(17,20)(24,28)(26,27), (5,6,7,8)(9,10)(11,12)(15,19)(17,20)(21,22)(23,25)(24,27)(26,28), (6,8)(10,12)(22,23)(24,28), (5,6)(7,8)(9,10,11,12)(13,14,16,18)(15,17,19,20)(21,23,25,22)(24,27,28,26), (3,4), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12)(13,15,16,19)(14,17,18,20)(21,24,25,28)(22,26,23,27));