Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C8 · Schur multiplier (H2) = C22×C42×C8 |
| direct product | C4×#1450 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C8 | C22 |
| commutator | C2 | C2×C42×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C42 |
| prime radical | C24 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,6,7,9,8,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(15,21)(18,24)(22,27)(25,28)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 87983);