Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C22×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C8×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(5,6,10,16,11,15,28,34,12,17,29,19,9,25,27,35)(7,18,21,23,20,13,30,24,14,31,33,22,26,32,36,8);(5,7)(6,13)(8,19)(9,26)(10,21)(11,20)(12,14)(15,31)(16,24)(17,32)(18,25)(22,34)(23,35)(27,36)(28,30)(29,33);(5,8,10,18,11,23,28,13,12,24,29,31,9,22,27,32)(6,14,16,33,15,26,34,36,17,7,19,21,25,20,35,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529778);