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×#26652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2303 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)(11,12,17,32,16,22,15,21)(13,23,31,18,27,38,26,34)(14,19,29,20,30,37,28,36)(24,39,25,33,41,35,40,42);(11,13)(12,18)(14,25)(15,31)(16,27)(17,26)(19,35)(20,33)(21,38)(22,34)(23,32)(24,29)(28,41)(30,40)(36,42)(37,39);(4,7)(8,10)(11,14)(12,19)(13,24)(15,28)(16,30)(17,29)(18,33)(20,32)(21,36)(22,37)(23,39)(25,31)(26,40)(27,41)(34,42)(35,38);(3,5,6,9)(4,8,7,10)(11,14)(12,20)(13,25)(15,29)(16,30)(17,28)(18,33)(19,32)(21,37)(22,36)(23,35)(24,31)(26,41)(27,40)(34,42)(38,39)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346794);