Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#11471 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C23 | C2×#99 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,9,7,10,5,8)(11,12)(13,17,18,15)(14,16);(12,15)(16,17);(3,5,7,6)(4,8,10,9)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#129599, #72240324, #72240326, … |
| maximal normal subgroup of | C2×#129599, #72240324, #72240326, … |
| maximal quotient of | #72261256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387284);