Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C4 · Schur multiplier (H2) = C228 |
| direct product | C24×#2380 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator | C22 | C27×C4 |
| Frattini, cosocle | C23 | C28 |
| prime radical | C25×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2255208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 8 generators on 28 points:
(15,19)(18,20)(21,22)(23,27)(24,25)(26,28);(14,17)(15,19)(22,25)(23,28);(3,4);(5,6);(9,10,11,12)(13,14,16,17)(15,20,19,18)(21,22)(23,26)(24,25)(27,28);(7,8);(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,6)(7,8)608 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((15,19)(18,20)(21,22)(23,27)(24,25)(26,28), (14,17)(15,19)(22,25)(23,28), (3,4), (5,6), (9,10,11,12)(13,14,16,17)(15,20,19,18)(21,22)(23,26)(24,25)(27,28), (7,8), (13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27), (1,2)(3,4)(5,6)(7,8));