Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C228 |
| direct product | C2×D4×#34980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C27 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 22636 conjugacy classes of subgroups; only the 559723 normal subgroups are shown.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 22 points:
(8,10)(12,14)(15,16)(17,20)(18,19)(21,22);(4,6)(8,10)(11,12)(13,14)(16,19)(20,21);(12,14);(11,12)(13,14);(1,2);(8,10);(15,17)(16,20)(18,22)(19,21);(3,4)(5,6)(7,8)(9,10)(15,16)(17,21)(18,19)(20,22)640 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((8,10)(12,14)(15,16)(17,20)(18,19)(21,22), (4,6)(8,10)(11,12)(13,14)(16,19)(20,21), (12,14), (11,12)(13,14), (1,2), (8,10), (15,17)(16,20)(18,22)(19,21), (3,4)(5,6)(7,8)(9,10)(15,16)(17,21)(18,19)(20,22));