Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7218 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 50 points:
(1,2);(3,4)(5,11,13,17)(6,9)(7,10)(8,12,15,18)(14,16)(19,20)(21,31,33,45)(22,28)(23,38)(24,40)(25,30)(26,32,41,46)(27,36)(29,39)(34,49,47,43)(35,50,48,42)(37,44);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)(19,21,23,34,39,48,37,46,25,33,36,47,24,35,22,32)(20,26,28,42,40,49,27,31,30,41,44,50,29,43,38,45)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34447);