Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#59442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C23×#127 | C22 |
| 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)(7,11)(8,12)(9,10)(13,14)(15,16)(17,18)(19,20)(21,22,25,23)(24,27,28,26);(6,10)(7,12)(14,18)(15,20)(24,28)(26,27);(7,12)(8,11)(13,14)(15,16)(17,18)(19,20)(21,23,25,22)(24,27,28,26);(5,7,9,12)(6,11,10,8)(13,15)(14,19)(16,18)(17,20)(21,24)(22,26)(23,27)(25,28);(5,8,9,11)(6,7,10,12)(13,16,17,19)(14,20,18,15);(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)(7,11)(8,12)(9,10)(13,14)(15,16)(17,18)(19,20)(21,22,25,23)(24,27,28,26), (6,10)(7,12)(14,18)(15,20)(24,28)(26,27), (7,12)(8,11)(13,14)(15,16)(17,18)(19,20)(21,23,25,22)(24,27,28,26), (5,7,9,12)(6,11,10,8)(13,15)(14,19)(16,18)(17,20)(21,24)(22,26)(23,27)(25,28), (5,8,9,11)(6,7,10,12)(13,16,17,19)(14,20,18,15), (3,4), (1,2)(3,4));