Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C27×C42 |
| direct product | C4×#33329 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×C42 |
| commutator | C4 | C22×C43 |
| Frattini, cosocle | C22×C8 | C25 |
| socle | C22 | C23×#97 |
| prime radical | C22×#41 | C22×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
5963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,12)(8,20)(9,14)(10,13)(11,15)(16,24)(17,30)(18,25)(19,26)(21,33)(22,28)(23,27)(29,34)(31,36)(32,35);(7,16,18,17)(9,22)(12,24,25,30)(14,28)(19,31,32,29)(21,23)(26,36,35,34)(27,33);(5,7,9,19,10,16,23,29,11,18,22,32,8,17,21,31)(6,12,14,26,13,24,27,34,15,25,28,35,20,30,33,36);(5,8,11,10)(6,13,15,20)(7,17,18,16)(9,21,22,23)(12,24,25,30)(14,27,28,33)(19,31,32,29)(26,34,35,36);(1,2,3,4)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((5,6)(7,12)(8,20)(9,14)(10,13)(11,15)(16,24)(17,30)(18,25)(19,26)(21,33)(22,28)(23,27)(29,34)(31,36)(32,35), (7,16,18,17)(9,22)(12,24,25,30)(14,28)(19,31,32,29)(21,23)(26,36,35,34)(27,33), (5,7,9,19,10,16,23,29,11,18,22,32,8,17,21,31)(6,12,14,26,13,24,27,34,15,25,28,35,20,30,33,36), (5,8,11,10)(6,13,15,20)(7,17,18,16)(9,21,22,23)(12,24,25,30)(14,27,28,33)(19,31,32,29)(26,34,35,36), (1,2,3,4));