Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11892 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1082 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)(5,9,10,7)(6,8)(11,12,16,22,18,24,17,23)(13,25,28,35,30,40,29,39)(14,31,15,34,33,21,32,20)(19,26,36,27,38,42,37,41);(3,5)(4,7)(6,10)(8,9)(11,13,18,30)(12,19,24,38)(14,27,33,41)(15,26,32,42)(16,29,17,28)(20,25,34,40)(21,35,31,39)(22,37,23,36);(5,10)(7,9)(11,14,17,32,18,33,16,15)(12,20,22,31,24,34,23,21)(13,26,29,41,30,42,28,27)(19,35,36,40,38,39,37,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389283);