Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#3036 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3036 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3354 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);(6,8)(9,10)(11,13)(12,14)(15,16);(5,6,7,8);(10,13)(11,16)(14,15);(10,14)(13,15)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72256880, #72256997, #72257029, … |
| maximal normal subgroup of | #72256880, #72256997, #72257029, … |
| maximal quotient of | C4×#21503, #72264814, #72265210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280000);