Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,16,23,18,24,17,22)(13,25,30,42,31,41,29,37)(14,32,26,20,33,21,15,34)(19,36,38,28,40,27,39,35);(4,7)(8,9)(11,13,26,28,17,30,14,27,18,31,15,35,16,29,33,36)(12,19,21,41,23,39,34,42,24,40,32,25,22,38,20,37);(3,5,6,10)(4,8,7,9)(11,14,16,26,18,33,17,15)(12,20,22,32,24,34,23,21)(13,27,29,28,31,36,30,35)(19,37,38,25,40,42,39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402583);