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×#10613 |
| 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);(3,4,7,14,9,13,8,12)(5,11,15,10,6,17,16,18)(19,20,24,27)(21,28,26,23)(22,29,32,30)(25,33,34,31);(3,5,7,15,9,6,8,16)(4,10,12,11,13,18,14,17)(19,21,23,20,24,26,28,27)(22,30,31,25,32,29,33,34);(3,6,9,5)(4,11,13,17)(7,15,8,16)(10,14,18,12)(19,22,24,32)(20,25,27,34)(21,29,26,30)(23,33,28,31)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149557);