Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
| direct product | C2×#7226 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C4×M16 | C23 |
| socle | C22 | #1288 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,26)(6,17,18,31)(7,11)(8,12)(9,14)(10,19,24,32)(16,27,30,34)(20,23)(21,28)(22,29)(25,33);(3,5)(4,9,11,21,12,22,23,25)(6,14)(7,16)(8,15)(10,13,17,27,24,26,31,34)(18,29)(19,28)(20,30)(32,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34509);