Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C22×#1276 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | #95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2962 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)(7,9)(8,11)(10,12)(13,14,15,17,16,18,19,20);(6,9,11,10)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#7867, #72237306, #72238287, … |
| maximal normal subgroup of | C22×#7867, #72237306, #72238287, … |
| maximal quotient of | #72237307 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88090);