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×#19535 |
| 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 | |
1598 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);(3,4,8,11)(5,12,15,18)(6,10,7,16)(9,13,17,14)(19,20)(21,23)(22,29)(24,27)(25,30)(26,28)(31,34)(32,33);(3,5,6,13,8,15,7,14)(4,9,16,18,11,17,10,12)(19,21,28,27,24,26,23,20)(22,29,30,34,32,33,31,25);(3,4)(5,9)(6,16)(7,10)(8,11)(12,14)(13,18)(15,17)(19,22,24,32)(20,25,27,34)(21,29,26,33)(23,31,28,30);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(19,23,24,28)(20,21,27,26)(22,30,32,31)(25,33,34,29)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261816);