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×#13439 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D16 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1126 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);(3,4)(5,6)(7,8)(9,20)(10,14)(11,26)(12,29)(13,19)(15,21)(16,24)(17,27)(18,25)(22,37)(23,33)(28,32)(30,35)(31,38)(34,36);(8,14)(9,21)(10,24)(11,27)(12,25)(16,18)(17,29)(19,26)(20,30)(22,38)(23,28)(31,34)(33,37)(35,36);(4,6)(7,9,10,21,25,35,27,38,13,23,11,28,12,22,24,36)(8,15,16,20,29,34,26,37,19,32,17,33,18,31,14,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406351);