Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17796 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2248 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);(4,7)(5,9)(8,10)(11,12,13,16,14,15,17,18);(3,4,5,7,6,8,9,10);(12,15)(16,18);(4,8)(7,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×M16×AGL(1,ℤ/8), C2×#6354077, C2×#6354085 |
| maximal normal subgroup of | C2×M16×AGL(1,ℤ/8), C2×#6354077, C2×#6354085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251513);