Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C225 |
| direct product | D4×#612424 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C27 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 15610 conjugacy classes of subgroups; only the 82091 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 20 points:
(10,12)(13,14)(15,16)(18,20);(2,4)(6,8)(10,12)(14,16)(17,18)(19,20);(18,20);(17,18)(19,20);(10,12);(6,8);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)580 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)(18,20), (2,4)(6,8)(10,12)(14,16)(17,18)(19,20), (18,20), (17,18)(19,20), (10,12), (6,8), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16));