Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#11640 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,17)(15,18)(19,20,22,27,23,28,32,40,24,26,33,38,34,39,44,48)(21,25,30,35,31,36,43,45,29,37,41,46,42,47,49,50);(3,5,8,15)(4,9,11,17)(6,14,7,13)(10,18,16,12)(19,21)(20,25)(22,30)(23,31)(24,29)(26,37)(27,35)(28,36)(32,43)(33,41)(34,42)(38,46)(39,47)(40,45)(44,49)(48,50);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(20,26)(21,29)(27,38)(28,39)(30,41)(31,42)(40,48)(43,49)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388333);