Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C228 |
| direct product | C22×#72233705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 8682 conjugacy classes of subgroups; only the 527483 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 28 points:
(10,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28);(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,27)(25,28);(6,8)(10,12)(21,22)(23,24)(25,27)(26,28);(6,8)(22,27)(24,26);(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27);(3,4);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12)(13,16)(14,19)(15,20)(17,18)(21,24)(22,28)(23,27)(25,26)616 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((10,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28), (13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,27)(25,28), (6,8)(10,12)(21,22)(23,24)(25,27)(26,28), (6,8)(22,27)(24,26), (10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27), (3,4), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12)(13,16)(14,19)(15,20)(17,18)(21,24)(22,28)(23,27)(25,26));