Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C22×#2010 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
8134 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)(8,16)(9,12)(10,14)(11,18)(15,20)(17,19);(5,7)(6,10)(8,17)(9,15)(11,20)(12,19)(13,16)(14,18);(10,19)(13,20)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338937);