Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1948 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
668 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);(3,4,8,15,10,16,9,14)(5,17,21,31,22,28,20,34)(6,23,24,12,25,13,7,26)(11,27,29,19,30,33,32,18);(3,5)(4,11,16,30)(6,18)(7,27)(8,20)(9,21)(10,22)(12,28,26,17)(13,31,23,34)(14,29,15,32)(19,25)(24,33);(3,6,8,24,10,25,9,7)(4,12,14,23,16,26,15,13)(5,18,20,33,22,19,21,27)(11,28,29,34,30,17,32,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402275);