Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7186 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C24 | C22⋊C8 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 46 points:
(1,2);(7,8)(9,13,14,11)(10,12)(15,16,19,24,20,25,34,42,21,26,35,43,18,23,32,40)(17,27,29,44,28,38,45,41,30,39,33,46,31,22,36,37);(3,4,5,6)(7,9)(8,11)(10,14)(12,13)(15,17)(16,22)(18,31)(19,33)(20,28)(21,30)(23,39)(24,41)(25,27)(26,38)(29,35)(32,45)(34,36)(37,43)(40,44)(42,46)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34100);