Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1280 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C22 | C42×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4);(9,10)(11,15,16,13)(12,14)(17,18,19,21,20,22,23,24);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12490);