Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62437 |
| 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 | |
6840 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)(11,12,15,19)(13,21,23,17)(14,25,27,41)(16,29,26,20)(18,32,33,28)(22,36,38,40)(24,31,37,35)(30,39,34,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,22)(15,23)(16,24)(18,30)(19,21)(20,31)(25,40)(26,37)(27,38)(28,39)(29,35)(32,42)(33,34)(36,41);(11,14,26,32,15,27,16,28)(12,18,29,41,19,33,20,25)(13,22,37,42,23,38,24,39)(17,30,35,36,21,34,31,40);(4,7)(8,10)(13,23)(17,21)(22,38)(24,37)(30,34)(31,35)(36,40)(39,42);(3,5,6,9)(4,8,7,10)(12,19)(13,23)(18,33)(20,29)(22,38)(24,37)(25,41)(39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486399);