Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C4×#100 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,14,12,13,15,16);(10,13)(11,15)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#1911, C2×#6443328, C2×#361625, … |
| maximal normal subgroup of | C2×C4×#1911, C2×#6443328, C2×#361625, … |
| maximal quotient of | C2×#182122 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277016);