Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C230 |
| direct product | C2×#72334301 |
| 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 6236 conjugacy classes of subgroups; only the 768203 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)(18,21)(19,22);(15,17)(16,20)(18,22)(19,21);(4,6)(8,10)(12,14)(17,21)(18,20);(4,6)(8,10)(15,17)(16,18)(19,21)(20,22);(8,10)(12,14);(1,2);(8,10);(3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,18)(16,21)(17,22)(19,20)760 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)(18,21)(19,22), (15,17)(16,20)(18,22)(19,21), (4,6)(8,10)(12,14)(17,21)(18,20), (4,6)(8,10)(15,17)(16,18)(19,21)(20,22), (8,10)(12,14), (1,2), (8,10), (3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,18)(16,21)(17,22)(19,20));