Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7195 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,11,12,8)(6,10)(7,9)(13,17,18,15)(14,16)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34);(3,5,6,13,7,12,14,18)(4,8,10,15,9,11,16,17)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34221);