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×#19547 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1572 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);(3,4,8,12)(5,9,14,18)(6,11,7,16)(10,15,17,13)(19,20)(21,23)(22,30)(24,29)(25,27)(26,32)(28,33)(31,34);(3,5)(4,9)(6,15)(7,13)(8,14)(10,16)(11,17)(12,18)(19,21,23,20,25,29,24,27)(22,28,32,26,33,30,31,34);(3,5)(4,10)(6,13)(7,15)(8,14)(9,16)(11,18)(12,17)(19,22)(20,26)(21,28)(23,32)(24,31)(25,33)(27,34)(29,30);(3,6,8,7)(4,11,12,16)(5,13,14,15)(9,10,18,17)(19,23,25,24)(20,21,27,29)(22,31,33,32)(26,30,34,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261830);