Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22401 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(7,8)(9,10)(11,12,14,16,13,15,17,18);(3,4)(5,6)(12,15)(16,18);(8,10);(4,6)(12,15)(16,18)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#407, C2×#6573582, C2×#6573573, … |
| maximal normal subgroup of | C2×D4×#407, C2×#6573582, C2×#6573573, … |
| maximal quotient of | C2×#359424, #72305921, #72306186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286180);