Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10611 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10614 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #126 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,12,8,13,7,17)(5,14,16,18,6,11,15,10)(19,20)(21,23)(22,30)(24,27)(25,28)(26,32)(29,33)(31,34);(3,5,9,16,8,6,7,15)(4,10,17,11,13,18,12,14)(19,21,24,28,25,27,23,20)(22,29,32,26,33,30,31,34);(3,6,8,5)(4,11,13,14)(7,15,9,16)(10,12,18,17)(19,22,25,33)(20,26,28,34)(21,29,27,30)(23,32,24,31)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149558);