Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#2231 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#476 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,11,16)(7,17,20,15)(8,21)(9,25,18,14)(10,12,27,29)(13,30)(19,33)(22,35)(23,34)(24,28)(26,31)(32,36);(5,7,18,27,11,20,9,10)(6,12,25,15,16,29,14,17)(8,19,23,36,24,30,22,26)(13,28,32,34,33,21,31,35);(5,8)(6,13)(7,19)(9,22)(10,26)(11,24)(12,28)(14,31)(15,34)(16,33)(17,35)(18,23)(20,30)(21,29)(25,32)(27,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403194);