Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22995 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#459 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,5)(7,8,10,9)(11,12)(13,15)(14,17)(16,18);(3,5,6,4)(8,9);(12,15)(13,18)(16,17);(7,9,10,8)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72280295, C2×C4×D4×D8, C2×#7521049, … |
| maximal normal subgroup of | #72280295, C2×C4×D4×D8, C2×#7521049, … |
| maximal quotient of | C2×#498306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289024);