Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C210 · Schur multiplier (H2) = C244 |
| direct product | C24×D4×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator, Frattini, cosocle | C22 | C210 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
475073979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 10 generators on 20 points:
(14,17)(18,20);(15,19)(18,20);(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,19)(17,20);(3,4);(5,6);(10,12);(7,8);(1,2)(3,4)(5,6)(7,8);(9,10,11,12)1360 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((14,17)(18,20), (15,19)(18,20), (13,14)(15,18)(16,17)(19,20), (13,15)(14,18)(16,19)(17,20), (3,4), (5,6), (10,12), (7,8), (1,2)(3,4)(5,6)(7,8), (9,10,11,12));