Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10645 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10646 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #126 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
558 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);(3,4,6,11,8,12,7,10)(5,9,14,17,15,16,13,18)(19,20)(21,29,32,42)(22,26)(23,27)(24,28)(25,31,39,36)(30,45,33,44)(34,40)(35,41)(37,43)(38,47,49,46)(48,50);(3,5)(4,9)(6,14)(7,13)(8,15)(10,18)(11,17)(12,16)(19,21,24,32)(20,25,28,39)(22,33,35,30)(23,36,37,31)(26,38,41,49)(27,42,43,29)(34,46,48,47)(40,45,50,44);(4,10)(5,13)(6,7)(9,16)(11,12)(14,15)(19,22,23,34,24,35,37,48)(20,26,27,40,28,41,43,50)(21,30,31,46,32,33,36,47)(25,38,29,44,39,49,42,45)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150285);