Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11632 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,19,15,20,24,32,16,18,25,30,26,31,36,40)(13,17,22,27,23,28,35,37,21,29,33,38,34,39,41,42);(4,8)(5,6)(9,10)(11,13)(12,17)(14,22)(15,23)(16,21)(18,29)(19,27)(20,28)(24,35)(25,33)(26,34)(30,38)(31,39)(32,37)(36,41)(40,42);(3,5,7,6)(4,9,10,8)(12,18)(13,21)(19,30)(20,31)(22,33)(23,34)(32,40)(35,41)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388311);