Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3217 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1876 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);(4,8)(6,13)(9,14)(10,15)(12,17)(16,18)(19,20)(21,29)(22,28)(23,25)(24,27)(26,30)(31,34)(32,33);(3,4,6,8,7,10,13,15)(5,9,12,14,11,16,17,18)(20,25)(26,33)(27,28)(29,34);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21)(20,26)(22,30)(23,31)(24,32)(25,33)(27,29)(28,34);(5,11)(9,16)(12,17)(14,18)(19,22,23,24)(20,27,25,28)(21,30,31,32)(26,29,33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261231);