Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13260 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1330 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);(4,6)(7,8,11,16,29,17,27,33,13,19,26,15,12,18,10,25)(9,20,21,35,22,30,37,32,24,36,38,31,23,34,28,14);(3,4,5,6)(7,9)(8,14)(10,21)(11,28)(12,22)(13,24)(15,30)(16,34)(17,31)(18,35)(19,32)(20,25)(23,29)(26,37)(27,38)(33,36);(7,10,12,26,13,27,29,11)(8,15,17,25,19,16,18,33)(9,21,22,37,24,38,23,28)(14,30,31,20,32,34,35,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402464);