Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8)(9,21)(10,25)(11,19)(12,28)(13,17)(14,20)(15,22)(16,26)(18,30)(23,34)(24,33)(27,37)(29,32)(31,36)(35,38);(3,4)(5,6)(7,9)(8,15)(10,11)(12,29)(13,22)(14,24)(16,28)(17,33)(18,34)(19,21)(20,25)(23,31)(26,30)(27,38)(32,37)(35,36);(4,6)(7,10,11,9,14,22,13,24)(8,16,17,32,20,35,19,34)(12,29,27,26,31,23,30,38)(15,18,21,36,25,37,33,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406226);