Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11458 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C22 | C2×#324 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,11,7,12,17,25,8,10,16,22,18,23,29,32)(5,9,14,20,15,21,28,31,13,19,26,24,27,30,34,33);(3,5)(4,9,10,19)(6,14)(7,15)(8,13)(11,24,22,20)(12,21,23,30)(16,26)(17,28)(18,27)(25,33,32,31)(29,34);(5,13)(9,19)(14,26)(15,27)(20,24)(21,30)(28,34)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387239);