Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60773 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7970 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)(11,12,16,20)(13,21,22,34)(14,18,25,32)(15,27,29,19)(17,28,30,24)(23,35,36,42)(26,39,38,33)(31,40,41,37);(3,5)(4,7)(6,10)(8,9)(13,22)(17,30)(21,34)(23,36)(24,28)(31,41)(35,42)(37,40);(5,10)(7,9)(11,13,16,22)(12,17,20,30)(14,23,25,36)(15,28,29,24)(18,31,32,41)(19,34,27,21)(26,40,38,37)(33,42,39,35);(4,8)(7,9)(11,14)(12,18)(13,23)(15,26)(16,25)(17,31)(19,33)(20,32)(21,35)(22,36)(24,37)(27,39)(28,40)(29,38)(30,41)(34,42);(14,25)(18,32)(23,36)(26,38)(31,41)(33,39)(35,42)(37,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482636);