Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3224 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)(6,17)(10,23)(11,21)(12,18)(14,29)(15,31)(16,28)(19,33)(22,24)(25,34)(26,30)(27,32);(3,4,7,12)(5,14,16,10)(6,18,17,9)(8,11,20,25)(13,26,28,24)(15,22,32,30)(19,34,33,21)(23,31,29,27);(6,19)(9,21)(15,27)(17,33)(18,34)(22,23)(29,30)(31,32);(3,5,7,16)(4,10,12,14)(6,15,17,32)(8,13,20,28)(9,22,18,30)(11,24,25,26)(19,27,33,31)(21,23,34,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254108);