Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1070 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)(5,10)(6,15)(7,17)(8,12)(9,13)(11,16)(14,18)(19,20,22,26,23,25,27,33)(21,24,29,31,28,32,30,34);(3,5,7,13,8,14,16,6)(4,9,11,10,12,15,17,18)(19,21,27,29,23,28,22,30)(20,24,26,34,25,32,33,31);(3,6,16,14,8,13,7,5)(4,10,17,9,12,18,11,15)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113065);