Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7147 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
418 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)(5,12,14,18)(6,10)(7,16)(8,11)(9,13,17,15)(19,20,22,26,23,28,33,40,24,27,34,39,35,41,46,49)(21,29,31,42,32,43,45,50,30,25,36,37,44,38,47,48);(3,5,7,15,8,14,6,13)(4,9,10,12,11,17,16,18)(19,21)(20,25,27,29)(22,31)(23,32)(24,30)(26,37,39,42)(28,38,41,43)(33,45)(34,36)(35,44)(40,48,49,50)(46,47)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33804);