Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10511 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,14)(5,15,16,18)(6,10,17,11)(7,13,8,12)(19,20)(21,24)(22,25)(23,27)(26,31)(28,32)(29,34)(30,33);(3,5)(4,10)(6,8)(7,17)(9,16)(11,14)(12,18)(13,15)(19,21,23,29)(20,24,27,34)(22,30,31,28)(25,33,26,32);(3,6,7,16,9,17,8,5)(4,11,12,18,14,10,13,15)(20,25)(22,31)(24,32)(26,27)(28,30)(33,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 148059);