Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#36901 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
67737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(6,9)(7,11)(13,14,17,20)(15,16,19,18)(21,22,26,28)(23,25,24,27);(5,6)(7,10)(8,9)(11,12)(15,19)(16,18)(21,22)(23,25)(24,27)(26,28);(6,9)(10,12)(13,15,17,19)(14,18,20,16)(21,23,26,24)(22,27,28,25);(3,4);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24,26,23)(22,25,28,27);(1,2)(3,4);(5,6)(7,12)(8,9)(10,11)(13,16)(14,19)(15,20)(17,18)(21,25)(22,24)(23,28)(26,27)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((6,9)(7,11)(13,14,17,20)(15,16,19,18)(21,22,26,28)(23,25,24,27), (5,6)(7,10)(8,9)(11,12)(15,19)(16,18)(21,22)(23,25)(24,27)(26,28), (6,9)(10,12)(13,15,17,19)(14,18,20,16)(21,23,26,24)(22,27,28,25), (3,4), (5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24,26,23)(22,25,28,27), (1,2)(3,4), (5,6)(7,12)(8,9)(10,11)(13,16)(14,19)(15,20)(17,18)(21,25)(22,24)(23,28)(26,27));