Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20260 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1779 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);(4,8)(6,15)(9,16)(10,11)(12,17)(13,18)(19,20)(21,25)(22,29)(23,32)(24,27)(26,28)(30,34)(31,33);(3,4,7,11)(5,12,14,9)(6,10,15,8)(13,16,18,17)(19,21,24,23)(20,25,27,32)(22,30,31,28)(26,29,34,33);(3,5,7,14)(4,9,11,12)(6,13,15,18)(8,16,10,17)(19,22)(20,26)(21,28)(23,30)(24,31)(25,33)(27,34)(29,32);(4,10)(6,15)(8,11)(9,17)(12,16)(13,18)(19,21,24,23)(20,25,27,32)(22,28,31,30)(26,33,34,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266812);