Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7198 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 50 points:
(1,2);(3,4)(5,11,12,8)(6,10)(7,9)(13,17,18,15)(14,16)(19,20,23,28,24,29,38,46,25,30,39,47,22,27,36,44)(21,31,33,48,32,42,49,45,34,43,37,50,35,26,40,41);(3,5,6,13,7,12,14,18)(4,8,10,15,9,11,16,17)(19,21)(20,26)(22,35)(23,37)(24,32)(25,34)(27,43)(28,45)(29,31)(30,42)(33,39)(36,49)(38,40)(41,47)(44,48)(46,50)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34230);