Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1379 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4);(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31);(5,6,7,8)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53500);