Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20172 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3391 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(12,17)(13,15)(18,20)(19,25)(21,24)(22,26);(4,8)(7,10)(11,12,16,20)(13,17,15,18)(14,22,23,25)(19,24,26,21);(3,4)(5,10)(6,8)(7,9)(11,13,16,15)(12,18,20,17)(14,21,23,24)(19,22,26,25);(11,14,16,23)(12,19,20,26)(13,21,15,24)(17,25,18,22)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266718);