Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#35207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
57657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,16)(17,18)(19,20);(5,7)(6,10)(8,11)(9,12)(14,18)(15,19)(22,25)(26,28);(5,7,9,12)(6,11,8,10)(13,15)(14,16)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,14)(15,20)(16,19)(17,18)(21,23,24,27)(22,26,25,28);(5,8,9,6)(7,11,12,10)(13,16,17,20)(14,15,18,19);(3,4);(1,2)(3,4)256 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((5,6,9,8)(7,11,12,10)(13,14)(15,16)(17,18)(19,20), (5,7)(6,10)(8,11)(9,12)(14,18)(15,19)(22,25)(26,28), (5,7,9,12)(6,11,8,10)(13,15)(14,16)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28), (5,7)(6,11)(8,10)(9,12)(13,14)(15,20)(16,19)(17,18)(21,23,24,27)(22,26,25,28), (5,8,9,6)(7,11,12,10)(13,16,17,20)(14,15,18,19), (3,4), (1,2)(3,4));