Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22772 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,12)(10,14)(11,13)(15,16)(17,23)(18,27)(19,21)(20,25)(22,28)(24,29)(26,30);(7,9,11,10)(8,12,13,14)(15,17)(16,20)(18,24)(19,26)(21,29)(22,30)(23,27)(25,28);(7,10,11,9)(8,12,13,14)(16,21)(20,29)(22,27)(23,30);(16,21)(17,24)(18,28)(20,23)(25,26)(29,30)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287686);