Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#8323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3254 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);(3,4,5,7,6,8,9,10)(11,12)(13,21)(14,25,27,18)(15,28)(16,20)(17,23)(19,30)(22,36,38,35)(24,34)(26,33,42,40)(29,32)(31,37,39,41);(11,13,16,24)(12,17,20,32)(14,22,27,38)(15,29,30,23)(18,31,25,39)(19,34,28,21)(26,41,42,37)(33,36,40,35);(11,14)(12,18)(13,22)(15,26)(16,27)(17,31)(19,33)(20,25)(21,35)(23,37)(24,38)(28,40)(29,41)(30,42)(32,39)(34,36)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92164);