Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8732 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #381 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1048 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)(5,12,14,8)(6,11)(7,10)(9,13,16,18)(15,17)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,5)(4,8)(6,13)(7,14)(9,11)(10,12)(15,18)(16,17)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(3,5)(4,9,10,16)(6,13)(7,14)(8,11,12,17)(15,18)(21,27)(24,30)(28,33)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97187);