Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#12994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,6)(7,8)(9,10)(11,12,16,21,31,40,29,38,17,22,15,20,14,19,30,39)(13,23,26,36,24,42,32,18,28,35,25,34,27,41,33,37);(3,4,5,6)(11,13)(12,18)(14,24)(15,32)(16,33)(17,28)(19,34)(20,41)(21,42)(22,37)(23,38)(25,29)(26,30)(27,31)(35,39)(36,40);(8,10)(11,14,17,31)(12,19,22,40)(13,24,28,27)(15,29,16,30)(18,34,37,36)(20,38,21,39)(23,42,35,41)(25,33,26,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401637);