Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C8×#360 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8);(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24)128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×AGL(1,ℤ/16), C2×C8×#360, #72287484, … |
| maximal normal subgroup of | C8×AGL(1,ℤ/16), C2×C8×#360, #72287484, … |
| maximal quotient of | C8×#1333, C2×C8×#360 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35866);