Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C225 |
| direct product | C22×#8387575 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 8919 conjugacy classes of subgroups; only the 419816 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 28 points:
(6,9)(7,11)(21,22,24,25)(23,27,28,26);(3,4);(6,9)(7,11)(14,17)(15,19)(22,25)(23,28);(1,2)(3,4);(6,9)(10,12)(14,17)(15,19)(23,28)(26,27);(14,17)(18,20)(23,28)(26,27);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23,24,28)(22,26,25,27);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,26)(24,25)(27,28)368 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((6,9)(7,11)(21,22,24,25)(23,27,28,26), (3,4), (6,9)(7,11)(14,17)(15,19)(22,25)(23,28), (1,2)(3,4), (6,9)(10,12)(14,17)(15,19)(23,28)(26,27), (14,17)(18,20)(23,28)(26,27), (5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23,24,28)(22,26,25,27), (5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,26)(24,25)(27,28));