Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#44656 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
63273 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)(7,8)(9,11)(10,12)(13,14,17,20)(15,18,16,19)(21,22,24,26)(23,25,28,27);(5,7)(6,10)(8,11)(9,12)(15,16)(18,19);(7,12)(8,10)(13,15,17,16)(14,18,20,19)(21,22,24,26)(23,25,28,27);(5,7)(6,8)(9,12)(10,11)(13,16,17,15)(14,19,20,18)(21,23)(22,25)(24,28)(26,27);(5,8)(6,7)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,26)(25,28);(3,4);(1,2)(3,4)232 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)(7,8)(9,11)(10,12)(13,14,17,20)(15,18,16,19)(21,22,24,26)(23,25,28,27), (5,7)(6,10)(8,11)(9,12)(15,16)(18,19), (7,12)(8,10)(13,15,17,16)(14,18,20,19)(21,22,24,26)(23,25,28,27), (5,7)(6,8)(9,12)(10,11)(13,16,17,15)(14,19,20,18)(21,23)(22,25)(24,28)(26,27), (5,8)(6,7)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,26)(25,28), (3,4), (1,2)(3,4));