Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7271 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #333 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,17,19,30)(10,22)(11,14)(12,26)(13,18,27,21)(15,25)(16,23)(20,28,34,38)(24,31)(29,36)(32,35,37,33);(7,9,10,20,23,21,36,35,11,19,24,34,12,18,25,33)(8,13,15,28,26,17,31,32,14,27,29,38,16,30,22,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34895);