Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13639 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C16 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | C22×C16 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,13,17,14,18,21,27,15,19,22,28,24,30,33,37,16,20,23,29,25,31,34,38,26,32,35,39,36,40,41,42);(4,7)(8,10);(3,5,6,9)(4,8,7,10)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419921);