Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226×C4 |
| direct product | C24×#2631 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2164808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 8 generators on 28 points:
(10,12)(13,14)(15,17)(16,18)(19,20)(22,23)(26,27);(10,12)(14,15)(18,19)(21,22,25,23)(24,26,28,27);(3,4);(5,6);(9,10,11,12)(13,15,17,14)(16,18,20,19)(21,23,25,22)(24,26,28,27);(7,8);(13,16)(14,18)(15,19)(17,20)(21,24,25,28)(22,26,23,27);(1,2)(3,4)(5,6)(7,8)512 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((10,12)(13,14)(15,17)(16,18)(19,20)(22,23)(26,27), (10,12)(14,15)(18,19)(21,22,25,23)(24,26,28,27), (3,4), (5,6), (9,10,11,12)(13,15,17,14)(16,18,20,19)(21,23,25,22)(24,26,28,27), (7,8), (13,16)(14,18)(15,19)(17,20)(21,24,25,28)(22,26,23,27), (1,2)(3,4)(5,6)(7,8));