Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11310 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| 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)(5,15,17,30,18,10,23,25)(6,11,22,29,20,27,21,28)(16,31,19,34,32,24,33,26);(3,5)(4,10)(6,19,20,33)(7,17)(8,23)(9,18)(11,26,27,34)(12,15)(13,25)(14,30)(16,21,32,22)(24,28,31,29);(3,6)(4,11,12,27)(5,16)(7,22)(8,21)(9,20)(10,24,15,31)(13,29,14,28)(17,33)(18,32)(19,23)(25,34,30,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265545);