Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,14,34,15,21,32,20)(9,22,27,35,10,28,30,37)(11,31,23,18,12,19,13,33)(16,29,38,25,17,26,36,24);(3,4)(5,6)(7,9,23,29,14,27,12,25,15,10,13,26,32,30,11,24)(8,16,31,37,20,36,33,28,21,17,19,35,34,38,18,22);(4,6)(7,10,15,9)(8,17,21,16)(11,26,12,29)(13,24,23,25)(14,27,32,30)(18,35,33,37)(19,22,31,28)(20,36,34,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406508);