Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#42606 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
72121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,16,19,20);(7,11)(10,12)(14,18)(15,19)(21,22)(23,26)(24,27)(25,28);(6,9)(7,11)(21,23)(22,25)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,26)(24,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,18)(17,20)(21,22)(23,25)(24,27)(26,28);(3,4);(1,2)(3,4)208 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,8,9)(7,10,11,12)(13,14,17,18)(15,16,19,20), (7,11)(10,12)(14,18)(15,19)(21,22)(23,26)(24,27)(25,28), (6,9)(7,11)(21,23)(22,25)(24,28)(26,27), (5,7,8,11)(6,10,9,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,26)(24,28)(25,27), (5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,18)(17,20)(21,22)(23,25)(24,27)(26,28), (3,4), (1,2)(3,4));