Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C2×#10586097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C27 |
| commutator, Frattini, cosocle | C22 | C28 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
829473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 8 generators on 22 points:
(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,22);(3,4)(5,6)(12,20)(16,22)(17,18)(19,21);(7,9)(8,14)(10,13)(11,15)(12,18)(16,19)(17,20)(21,22);(8,15)(9,10)(12,20)(19,21);(7,10,13,9)(8,11,15,14)(12,17,20,18)(16,21,22,19);(7,11)(8,10)(9,15)(12,19)(13,14)(16,18)(17,22)(20,21);(7,12)(8,16)(9,17)(10,18)(11,19)(13,20)(14,21)(15,22);(1,2)328 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((4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,22), (3,4)(5,6)(12,20)(16,22)(17,18)(19,21), (7,9)(8,14)(10,13)(11,15)(12,18)(16,19)(17,20)(21,22), (8,15)(9,10)(12,20)(19,21), (7,10,13,9)(8,11,15,14)(12,17,20,18)(16,21,22,19), (7,11)(8,10)(9,15)(12,19)(13,14)(16,18)(17,22)(20,21), (7,12)(8,16)(9,17)(10,18)(11,19)(13,20)(14,21)(15,22), (1,2));