Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1377 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C22 | C42×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
802 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);(9,10,11,15,12,16,18,22,13,14,17,20,19,21,23,24);(5,6,7,8)(10,14)(11,17)(16,21)(18,23)176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#13025, C4×#13138, C4×#12659 |
| maximal normal subgroup of | C4×#13025, C4×#13138, C4×#12659 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53497);