Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | D4×#334 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4);(2,4);(6,9)(7,11)(10,12)(13,14,15,17,16,18,19,20);(5,6,7,9,8,10,11,12)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72216394, D4×#1580, #72216483, … |
| maximal normal subgroup of | #72216394, D4×#1580, #72216483, … |
| maximal quotient of | #72216392, #72216394, #72225416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132582);