Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C221 |
| direct product | C22×#612618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C29 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2798184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(13,14,17,18)(15,19,20,16)(21,22,25,23)(24,26,28,27);(5,6,7,8)(10,12)(14,18)(16,19)(21,23,25,22)(24,26,28,27);(9,10)(11,12)(13,15)(14,16)(17,20)(18,19);(3,4);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12);(13,16)(14,15)(17,19)(18,20)(21,24)(22,26)(23,27)(25,28)464 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((13,14,17,18)(15,19,20,16)(21,22,25,23)(24,26,28,27), (5,6,7,8)(10,12)(14,18)(16,19)(21,23,25,22)(24,26,28,27), (9,10)(11,12)(13,15)(14,16)(17,20)(18,19), (3,4), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12), (13,16)(14,15)(17,19)(18,20)(21,24)(22,26)(23,27)(25,28));