Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C228 |
| direct product | C2×#72323292 |
| 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 32440 conjugacy classes of subgroups; only the 555179 normal subgroups are shown.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 22 points:
(12,14);(11,12)(13,14)(16,20)(18,21);(15,16)(17,21)(18,22)(19,20);(1,2);(4,6)(8,10)(17,22)(18,21);(17,22)(18,21);(7,8)(9,10)(15,17,19,22)(16,21,20,18);(3,4)(5,6)(7,8)(9,10)(15,18,19,21)(16,22,20,17)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((12,14), (11,12)(13,14)(16,20)(18,21), (15,16)(17,21)(18,22)(19,20), (1,2), (4,6)(8,10)(17,22)(18,21), (17,22)(18,21), (7,8)(9,10)(15,17,19,22)(16,21,20,18), (3,4)(5,6)(7,8)(9,10)(15,18,19,21)(16,22,20,17));