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) = C28 |
| direct product | C2×#18087 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,10)(6,12)(7,11)(8,13)(9,14)(15,18)(16,17)(19,20,24,27,25,21,23,28)(22,26,31,33,32,29,30,34);(3,5)(4,9)(6,14)(7,16)(8,15)(10,11)(12,18)(13,17)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34);(3,5)(4,10)(6,16)(7,14)(8,15)(9,11)(12,17)(13,18)(19,22,24,31,25,32,23,30)(20,26,27,33,21,29,28,34);(3,6,8,7)(4,11,13,12)(5,14,15,16)(9,10,17,18)(19,23,25,24)(20,27,21,28)(22,30,32,31)(26,33,29,34)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252327);