Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10227 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,8)(5,9)(6,10)(11,12,17,22)(13,23,26,34)(14,19,30,37)(15,21,16,20)(18,27,33,25)(24,39,40,35)(28,42,41,32)(29,38,31,36);(3,4,7,5)(6,9,10,8)(11,13,17,26)(12,18,22,33)(14,28,30,41)(15,27,16,25)(19,35,37,39)(20,34,21,23)(24,31,40,29)(32,38,42,36);(3,5,7,4)(6,8,10,9)(11,14,17,30)(12,19,22,37)(13,24,26,40)(15,29,16,31)(18,32,33,42)(20,36,21,38)(23,35,34,39)(25,28,27,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141024);