Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C4×#440 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8,10,9)(11,12)(13,15)(14,16)(17,18);(12,15)(13,18)(16,17);(7,9,10,8)(12,16)(15,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#440, C2×C4×#3114, C2×#6444787, … |
| maximal normal subgroup of | C2×D4×#440, C2×C4×#3114, C2×#6444787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277252);