Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13577 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| socle | C22 | C2×#353 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1126 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)(5,15,17,31)(6,11)(7,21)(8,13,24,28)(9,14)(10,16,26,33)(12,23,29,20)(18,25,19,32)(22,30,34,27);(3,5)(4,10)(6,16)(7,22)(8,17)(9,19)(11,25)(12,15)(13,30)(14,27)(18,24)(20,33)(21,31)(23,34)(26,28)(29,32);(3,6,9,7)(4,11,14,21)(5,16,19,22)(8,20,24,23)(10,25,27,31)(12,28,29,13)(15,26,32,30)(17,33,18,34)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419786);