Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13134 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
804 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);(4,5)(7,8,13,18,11,17,14,29)(9,19,22,36,21,15,12,27)(10,23,26,35,24,38,30,31)(16,28,33,20,32,37,34,25);(3,4,6,5)(7,9,14,12,11,21,13,22)(8,15,29,36,17,19,18,27)(10,20,30,28,24,25,26,37)(16,31,34,38,32,35,33,23);(3,5,6,4)(7,10,11,24)(8,16,17,32)(9,20,21,25)(12,28,22,37)(13,26,14,30)(15,31,19,35)(18,33,29,34)(23,36,38,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402060);