Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C22⋊C4×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,7,8);(2,4);(9,10,11,12)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72264246, #72264298, #72264306, … |
| maximal normal subgroup of | #72264246, #72264298, #72264306, … |
| maximal quotient of | #99×#102, #72314220, #72314266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263285);