Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33930 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(8,9)(11,12,16,20)(13,21,24,40)(14,25,29,39)(15,30,27,19)(17,32,34,22)(18,35,37,28)(23,41,31,36)(26,38,42,33);(3,4,6,7)(5,9,10,8);(11,13,16,24)(12,17,20,34)(14,26,29,42)(15,31,27,23)(18,36,37,41)(19,38,30,33)(21,39,40,25)(22,35,32,28);(3,5,6,10)(4,8,7,9)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531055);