Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33900 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,17,22)(13,23,24,40)(14,21,16,19)(15,28,31,36)(18,33,34,26)(20,27,38,30)(25,41,32,37)(29,39,42,35);(3,5)(4,7)(6,10)(8,9)(13,24)(18,34)(23,40)(25,32)(26,33)(29,42)(35,39)(37,41);(11,13,17,24)(12,18,22,34)(14,25,16,32)(15,29,31,42)(19,35,21,39)(20,37,38,41)(23,36,40,28)(26,27,33,30);(5,10)(7,9)(11,14,17,16)(12,19,22,21)(13,25,24,32)(15,27,31,30)(18,35,34,39)(20,36,38,28)(23,37,40,41)(26,29,33,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530908);