Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#13108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1917 |
| prime radical | C24 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
574 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);(3,4,8,15,9,13,27,32,10,14,28,16,7,23,25,19)(5,6,12,22,17,11,29,30,18,21,31,26,24,20,34,33);(3,5,8,12,9,17,27,29,10,18,28,31,7,24,25,34)(4,11,15,30,13,21,32,26,14,20,16,33,23,6,19,22);(3,6,7,20,10,21,9,11)(4,12,13,29,14,31,23,34)(5,16,17,19,18,15,24,32)(8,26,25,30,28,22,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401976);