Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17869 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2212 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);(3,4)(5,7)(6,8)(9,10)(11,12,14,13,15,16,18,17);(4,7)(5,10)(8,9);(11,13,14,16,15,17,18,12);(4,8)(7,9)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6360309, C2×#6360338, C2×#6360305, … |
| maximal normal subgroup of | C2×#6360309, C2×#6360338, C2×#6360305, … |
| maximal quotient of | C2×#193537, C2×#208674 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251740);