Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,20,17,23,29,22)(13,24,26,38,28,39,40,33)(14,21,16,19,30,36,31,37)(18,27,32,41,35,42,34,25);(3,5)(4,7)(6,10)(8,9)(11,13,16,27,15,26,30,41,17,28,31,42,29,40,14,25)(12,18,21,33,22,34,37,39,23,35,36,38,20,32,19,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34520);