Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#1501 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
5650 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)(7,11,12,9)(8,10)(14,18,19,20);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,20)(17,19);(13,15,17,16)(14,18,19,20)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250481);