Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C8×#100 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1490 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);(3,4,5,7,6,8,9,10);(11,12,13,16,14,15,17,18);(12,15)(13,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#211042, C2×#211082, C2×#182122 |
| maximal normal subgroup of | C2×#211042, C2×#211082, C2×#182122 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89288);