Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9778 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1022 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)(5,11)(6,14)(7,9)(8,13)(10,15)(12,17)(16,18)(19,20,24,28,25,27,23,21)(22,30,32,34,33,29,31,26);(3,5,6,13,7,12,15,18)(4,8,10,11,9,16,14,17)(19,21,24,27,25,28,23,20)(22,30,32,26,33,29,31,34);(4,9)(5,12)(10,14)(13,18)(19,22,23,31,25,33,24,32)(20,26,28,30,27,34,21,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112851);