Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10208 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(5,12)(6,7)(9,13)(10,16)(14,15)(17,18)(19,20)(21,31)(22,36)(23,29)(24,28)(25,30)(26,33)(27,37)(32,43)(34,42)(35,41)(38,45)(39,44)(40,48)(46,47)(49,50);(3,4)(5,13)(6,10)(7,16)(8,11)(9,15)(12,17)(14,18)(19,21,25,35)(20,26,30,42)(22,32,38,48)(23,34,24,33)(27,43,44,40)(28,41,29,31)(36,46,45,50)(37,49,39,47);(3,5)(4,9)(6,15)(7,12)(8,14)(10,18)(11,17)(13,16)(19,22,25,38)(20,27,30,44)(21,32,35,48)(23,37,24,39)(26,40,42,43)(28,36,29,45)(31,46,41,50)(33,47,34,49)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140012);