Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 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)(7,8,12,13,14,18,11,17)(9,19,22,15,23,32,21,33)(10,24,26,27,29,30,28,16)(20,36,37,25,34,31,38,35);(7,9,14,23)(8,15,18,33)(10,25,29,35)(11,22,12,21)(13,19,17,32)(16,34,27,20)(24,37,30,38)(26,36,28,31);(4,6)(7,10,11,26,14,29,12,28)(8,16,17,24,18,27,13,30)(9,20,21,37,23,34,22,38)(15,31,19,35,33,36,32,25)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406556);