Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,17,21)(13,18,24,34)(14,25,26,39)(15,20,16,30)(19,27,35,28)(22,36,37,42)(23,32,29,33)(31,40,41,38);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,29)(16,23)(17,24)(19,31)(20,33)(21,34)(25,36)(26,37)(27,40)(28,38)(30,32)(35,41)(39,42);(11,14,16,28,17,26,15,27)(12,19,20,25,21,35,30,39)(13,22,23,38,24,37,29,40)(18,31,33,36,34,41,32,42);(5,10)(7,9)(14,26)(19,35)(22,37)(25,39)(27,28)(31,41)(36,42)(38,40);(4,8)(7,9)(11,15,17,16)(12,20,21,30)(13,23,24,29)(14,27,26,28)(18,32,34,33)(19,25,35,39)(22,38,37,40)(31,42,41,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484783);