Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C213 |
| direct product | C22×#1786 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25×D4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
30338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,8)(9,10,11,12)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19);(6,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#21452, #72239786, #72239789, … |
| maximal normal subgroup of | C22×#21452, #72239786, #72239789, … |
| maximal quotient of | C22×#12736, #72265545, C2×#185958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277903);