Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,7,8,9)(11,12,16,20,17,21,15,31)(13,22,26,33,25,37,24,18)(14,27,29,35,30,19,28,34)(23,36,40,32,39,42,38,41);(3,4,5,9,6,8,10,7)(11,13,17,25)(12,18)(14,23,30,39)(15,24,16,26)(19,32)(20,22)(21,33)(27,41)(28,38,29,40)(31,37)(34,42)(35,36);(11,14,17,30)(12,19,21,27)(13,23,25,39)(15,29,16,28)(18,32,33,41)(20,35,31,34)(22,36,37,42)(24,40,26,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104559);