Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#59385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
71369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,9,10)(7,8,12,11)(16,20)(18,19)(22,25)(23,27);(6,10)(7,12)(13,14,17,15)(16,19,20,18)(21,22,24,25)(23,28,27,26);(7,12)(8,11);(3,4);(5,7,9,12)(6,11,10,8)(13,15,17,14)(16,18,20,19)(21,23,24,27)(22,26,25,28);(1,2)(3,4);(5,8,9,11)(6,12,10,7)(13,16)(14,18)(15,19)(17,20)(21,22)(23,28)(24,25)(26,27)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((5,6,9,10)(7,8,12,11)(16,20)(18,19)(22,25)(23,27), (6,10)(7,12)(13,14,17,15)(16,19,20,18)(21,22,24,25)(23,28,27,26), (7,12)(8,11), (3,4), (5,7,9,12)(6,11,10,8)(13,15,17,14)(16,18,20,19)(21,23,24,27)(22,26,25,28), (1,2)(3,4), (5,8,9,11)(6,12,10,7)(13,16)(14,18)(15,19)(17,20)(21,22)(23,28)(24,25)(26,27));