Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20117 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1600 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);(3,4,8,6)(5,11,13,18)(7,15,17,10)(9,16,14,12)(19,20,24,27)(21,25,23,32)(22,29,31,33)(26,30,34,28);(3,5,8,13)(4,9,6,14)(7,16,17,12)(10,18,15,11)(19,21,24,23)(20,25,27,32)(22,30,31,28)(26,29,34,33);(19,22,24,31)(20,26,27,34)(21,28,23,30)(25,33,32,29);(3,6,8,4)(5,11,13,18)(7,10,17,15)(9,16,14,12)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266644);