Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1220 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);(3,4,7,11,8,12,6,9)(5,13,15,10,16,17,14,18)(19,20,23,28)(21,29)(22,25)(24,27,32,34)(26,30)(31,33);(4,9,12,11)(10,13,18,17)(19,21,23,22)(20,25)(24,31,32,30)(26,27)(28,29)(33,34);(3,5)(4,10)(6,14)(7,15)(8,16)(9,13)(11,17)(12,18)(19,22,23,21)(20,26,28,33)(24,30,32,31)(25,27,29,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108230);