Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#18359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2631 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,18)(10,23)(11,21)(12,19)(14,29)(16,32)(17,28)(20,31)(22,25)(24,34)(26,30)(27,33);(3,4,7,12)(5,14,17,10)(6,19,18,9)(8,11,15,24)(13,26,28,25)(16,22,33,30)(20,34,31,21)(23,32,29,27);(6,20)(9,21)(16,27)(18,31)(19,34)(22,23)(29,30)(32,33);(3,5,15,28)(4,10,24,26)(6,16,31,32)(7,17,8,13)(9,22,34,29)(11,25,12,14)(18,33,20,27)(19,30,21,23)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254116);