Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
| direct product | C4×#1964 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(6,8)(9,10)(11,13,15,16)(12,14);(9,11)(10,13,14,16)(12,15);(5,6,7,8)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264005, #72264012, C4×#24260, … |
| maximal normal subgroup of | #72264005, #72264012, C4×#24260, … |
| maximal quotient of | #72279972, #72281046, #72311871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261802);