Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#2104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #401 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,13,15,10)(8,11,17,20)(9,12)(14,19)(16,18);(5,7)(6,10)(8,16,17,14)(9,15)(11,19,20,18)(12,13);(5,8)(6,11,12,20)(7,14)(9,17)(10,18,13,19)(15,16)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340005);