Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6786 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);(4,7)(5,9)(11,12,16,20)(13,17,22,30)(14,24,25,39)(15,27,28,19)(18,32,33,26)(21,34,36,42)(23,38,37,31)(29,40,41,35);(3,4)(5,8)(6,7)(9,10)(11,13,16,22)(12,17,20,30)(14,21,25,36)(15,23,28,37)(18,29,33,41)(19,31,27,38)(24,34,39,42)(26,35,32,40);(11,14,15,26,16,25,28,32)(12,18,19,24,20,33,27,39)(13,21,23,35,22,36,37,40)(17,29,31,34,30,41,38,42);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(13,22)(14,25)(17,30)(18,33)(23,37)(24,39)(26,32)(31,38)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486414);