Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11302 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1443 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,14,9,12,7,13)(5,10,18,27,16,26,17,25)(6,19,21,28,22,29,23,11)(15,30,31,34,20,33,32,24);(3,5,7,18,9,16,8,17)(4,10,14,25,12,26,13,27)(6,20,21,31,22,15,23,32)(11,24,29,30,28,34,19,33);(3,6)(4,11,12,28)(5,15,16,20)(7,21)(8,23)(9,22)(10,24)(13,19,14,29)(17,31,18,32)(25,30)(26,34)(27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265530);