Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217 |
| direct product | C2×#72214076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#2403 |
| commutator, Frattini, cosocle | C23×C4 | C26 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
At least 12243 conjugacy classes of subgroups; only the 6512 normal subgroups are shown.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(9,15)(10,17)(12,18)(14,20)(16,21)(19,22);(12,19)(18,22);(3,4)(5,6)(12,19)(18,22);(4,6)(7,8,11,13)(9,12,16,19)(10,14,17,20)(15,18,21,22);(7,9,10,15,11,16,17,21)(8,12,14,18,13,19,20,22)224 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (9,15)(10,17)(12,18)(14,20)(16,21)(19,22), (12,19)(18,22), (3,4)(5,6)(12,19)(18,22), (4,6)(7,8,11,13)(9,12,16,19)(10,14,17,20)(15,18,21,22), (7,9,10,15,11,16,17,21)(8,12,14,18,13,19,20,22));