Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×D4×#100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(3,4,5,6);(4,6);(7,8,9,12,10,11,13,14);(8,11)(9,13)110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6385983, D4×#1783, #72240267, … |
| maximal normal subgroup of | C2×#6385983, D4×#1783, #72240267, … |
| maximal quotient of | D4×#1783, C2×#361625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286214);