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) = C29 |
| direct product | C2×D4×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6);(4,6);(7,8,11,14,12,15,10,20)(9,16,18,21,19,22,17,13);(7,9,10,17,12,19,11,18)(8,13,14,16,15,21,20,22)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#365, C2×D4×#444, C2×D4×#361, … |
| maximal normal subgroup of | C2×D4×#365, C2×D4×#444, C2×D4×#361, … |
| maximal quotient of | C2×D4×#334, C2×#360188, D4×#1810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287291);