Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,7)(11,12)(14,15)(16,17)(19,20,24,28)(21,29,31,34)(22,27,23,26)(25,32,33,30);(3,4,8,12)(5,10,15,18)(6,11,7,9)(13,17,14,16)(19,21,24,31)(20,25,28,33)(22,30,23,32)(26,29,27,34);(3,5)(4,10)(6,13)(7,14)(8,15)(9,16)(11,17)(12,18)(22,23)(26,27)(29,34)(30,32);(3,6,8,7)(4,11,12,9)(5,14,15,13)(10,16,18,17)(19,22,24,23)(20,26,28,27)(21,30,31,32)(25,29,33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258689);