Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#41049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
69473 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)(7,12)(8,11)(9,10)(14,16)(18,19)(21,22,25,26)(23,27,28,24);(6,10)(7,11)(13,14,17,16)(15,19,20,18);(15,20)(18,19)(21,23,25,28)(22,24,26,27);(3,4);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4);(5,8)(6,11)(7,10)(9,12)(13,16,17,14)(15,18,20,19)(21,24)(22,23)(25,27)(26,28)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((5,6)(7,12)(8,11)(9,10)(14,16)(18,19)(21,22,25,26)(23,27,28,24), (6,10)(7,11)(13,14,17,16)(15,19,20,18), (15,20)(18,19)(21,23,25,28)(22,24,26,27), (3,4), (5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27), (1,2)(3,4), (5,8)(6,11)(7,10)(9,12)(13,16,17,14)(15,18,20,19)(21,24)(22,23)(25,27)(26,28));