Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C8×#471 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×AGL(1,ℤ/8) | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8);(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118232);