Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C22×C8 |
| direct product | C2×#7052 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C82 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C82 | C23 |
| socle, prime radical | C23 | C82 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,5,9,6,10,11,16,7,8,12,14,13,15,17,18);(4,8)(9,14)(10,15)(16,18)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29400);