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×#18069 |
| 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×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1688 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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20,24,27,23,26,32,34)(21,28,30,31,29,25,33,22);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,21)(20,25)(22,27)(23,29)(24,30)(26,28)(31,34)(32,33);(19,22,24,28,23,31,32,25)(20,21,27,30,26,29,34,33);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(21,29)(22,31)(25,28)(30,33)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252305);