Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,11,17,25,33,26,35,13,15,10,16,12,18,28,37)(9,19,22,32,29,36,24,14,21,31,20,38,23,34,27,30);(3,4,5,6)(7,9)(8,14)(10,24)(11,27)(12,29)(13,21)(15,30)(16,34)(17,36)(18,38)(19,35)(20,26)(22,28)(23,25)(31,37)(32,33);(8,15)(10,11)(14,30)(18,33)(20,22)(24,27)(26,28)(32,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401657);