Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C16 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×Q8×C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C16 | C22 |
| commutator | C2 | C22×C4×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C23×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12);(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400449);