Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26650 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,20,16,21,31,35)(13,22,25,33,26,38,30,17)(14,18,27,36,29,37,28,19)(23,39,40,42,41,34,24,32);(11,13,16,26)(12,17,21,33)(14,24,29,40)(15,30,31,25)(18,34,37,39)(19,32,36,42)(20,38,35,22)(23,27,41,28);(4,7)(8,10)(11,14)(12,18)(13,23)(15,27)(16,29)(17,32)(19,35)(20,36)(21,37)(22,39)(24,30)(25,40)(26,41)(28,31)(33,42)(34,38);(3,5,6,9)(4,8,7,10)(11,14)(12,19)(13,24)(15,28)(16,29)(17,32)(18,35)(20,37)(21,36)(22,34)(23,30)(25,41)(26,40)(27,31)(33,42)(38,39)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346800);