Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,7)(11,12)(14,15)(16,17)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,4)(5,10)(6,11)(7,9)(8,12)(13,17)(14,18)(15,16)(22,31)(25,30)(27,34)(29,32);(3,5)(4,10)(6,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31);(3,6,8,7)(4,11,12,9)(5,13,14,15)(10,17,18,16)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258577);