Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8917 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C2×#121 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,12,9,13,7,15)(5,14,11,10,17,18,16,6)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,5,8,11,9,17,7,16)(4,10,12,18,13,6,15,14)(22,31)(25,30)(27,34)(29,32);(3,6)(4,11)(5,15)(7,18)(8,14)(9,10)(12,17)(13,16)(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100303);