Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1648 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);(3,4,7,11)(5,8,12,16)(6,9,14,10)(13,15,18,17)(20,24)(21,28)(22,25)(27,32)(29,30)(31,33);(3,5)(4,8)(6,13)(7,12)(9,15)(10,17)(11,16)(14,18)(19,20,25,32,23,27,22,24)(21,26,28,33,29,34,30,31);(4,9)(6,14)(8,15)(10,11)(13,18)(16,17)(19,21)(20,26)(22,30)(23,29)(24,31)(25,28)(27,34)(32,33);(5,12)(8,16)(13,18)(15,17)(20,27)(21,29)(24,32)(28,30)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261794);