Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9236 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #386 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9)(19,20)(21,25,22,30)(23,28)(24,27)(26,31,34,29)(32,33);(3,5)(4,9,11,17)(6,14)(7,15)(8,13)(10,12,16,18)(19,21)(20,25,27,30)(22,24)(23,31)(26,28,34,33)(29,32);(5,13)(9,17)(12,18)(14,15)(19,22,24,21)(20,26,27,34)(23,29,32,31)(25,33,30,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105215);