Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C221 |
| direct product | C22×#613325 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
928520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,8,7)(9,10)(11,12)(13,14,16,18)(15,17,19,20)(22,25)(26,27);(6,7)(9,10,11,12)(13,15,16,19)(14,17,18,20)(21,22,24,25)(23,27,28,26);(10,12)(15,19)(17,20);(5,7,8,6)(9,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,28)(22,26,25,27);(3,4);(1,2)(3,4);(13,14)(15,20)(16,18)(17,19)(21,22)(23,27)(24,25)(26,28)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((5,6,8,7)(9,10)(11,12)(13,14,16,18)(15,17,19,20)(22,25)(26,27), (6,7)(9,10,11,12)(13,15,16,19)(14,17,18,20)(21,22,24,25)(23,27,28,26), (10,12)(15,19)(17,20), (5,7,8,6)(9,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,28)(22,26,25,27), (3,4), (1,2)(3,4), (13,14)(15,20)(16,18)(17,19)(21,22)(23,27)(24,25)(26,28));