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×#17835 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1746 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);(3,4,9,6)(5,12,16,18)(7,11,8,17)(10,14,13,15)(19,20,23,27,24,28,22,26)(21,29,31,34,32,33,30,25);(3,5,8,15,9,16,7,14)(4,10,11,12,6,13,17,18)(19,21,22,30,24,32,23,31)(20,25,27,29,28,34,26,33);(3,6,9,4)(5,13,16,10)(7,11,8,17)(12,15,18,14)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29);(3,7,9,8)(4,11,6,17)(5,14,16,15)(10,12,13,18)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251575);