Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C210 |
| direct product | C2×#10964 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C25 | C22×C4 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
6918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(15,16,17,19,18,20,21,22);(4,6)(11,12)(13,14);(8,10)(12,14)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#393915, C2×#393909, C2×#357415, … |
| maximal normal subgroup of | C2×#393915, C2×#393909, C2×#357415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264610);