Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
| direct product | C2×#8164 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,7)(5,9)(11,12,15,17,14,13,16,18);(3,4,5,8,6,7,9,10)(12,13)(17,18);(11,13,15,18,14,12,16,17)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#211045, C2×#211085, C2×#182122 |
| maximal normal subgroup of | C2×#211045, C2×#211085, C2×#182122 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89300);