Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#40926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
68149 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)(9,10)(11,12)(21,22)(23,27)(24,28)(25,26);(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,28)(26,27);(6,10)(8,12)(14,16)(18,20)(22,26)(24,27);(3,4);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4);(5,8)(6,11)(7,10)(9,12)(13,16,17,14)(15,18,19,20)(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)(7,8)(9,10)(11,12)(21,22)(23,27)(24,28)(25,26), (13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,28)(26,27), (6,10)(8,12)(14,16)(18,20)(22,26)(24,27), (3,4), (5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,24)(25,26)(27,28), (1,2)(3,4), (5,8)(6,11)(7,10)(9,12)(13,16,17,14)(15,18,19,20)(21,24)(22,23)(25,27)(26,28));