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×#25985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2609 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);(3,4,6,8)(5,9,10,7)(11,12)(13,23)(14,18)(15,21)(16,22)(17,26)(19,28)(20,32)(24,37)(25,34)(27,35)(29,38)(30,39)(31,36)(33,40)(41,42);(11,13,16,27)(12,17,22,36)(14,25,30,40)(15,31,32,26)(18,37,39,42)(19,34,38,33)(20,35,21,23)(24,28,41,29);(3,5,6,10)(4,7,8,9)(11,14)(12,18)(13,24)(15,28)(16,30)(17,33)(19,21)(20,38)(22,39)(23,37)(25,31)(26,40)(27,41)(29,32)(34,36)(35,42);(11,14)(12,19)(13,25)(15,29)(16,30)(17,34)(18,21)(20,39)(22,38)(23,37)(24,31)(26,41)(27,40)(28,32)(33,36)(35,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345350);