Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9733 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(4,6)(7,8,9,12,10,13,14,11)(15,16)(17,22)(18,27)(19,29)(20,24)(21,25)(23,28)(26,30);(3,4,5,6)(8,11)(9,14)(12,13)(15,17,19,25,20,26,28,18)(16,21,23,22,24,27,29,30);(15,18,28,26,20,25,19,17)(16,22,29,21,24,30,23,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112363);