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×C4 |
| direct product | C22×#1478 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
6330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,15,17,16,18,19,20);(6,10)(7,8)(11,12);(5,7,9,8)(6,11,12,10)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#204054, C22×#17448, C2×#184843, … |
| maximal normal subgroup of | C2×#204054, C22×#17448, C2×#184843, … |
| maximal quotient of | C22×#8170, C2×#481116, C2×#184843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250097);