Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11939 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
784 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);(4,7)(5,9)(8,10)(11,12,16,21,14,19,29,20,17,22,32,37,30,38,15,31)(13,23,26,35,28,39,42,33,27,40,41,18,24,34,25,36);(3,4,6,8)(5,10,9,7)(11,13,17,27)(12,18,22,35)(14,24,30,28)(15,26,29,41)(16,25,32,42)(19,33,38,36)(20,39,31,34)(21,40,37,23);(11,14,17,30)(12,19,22,38)(13,24,27,28)(15,16,29,32)(18,33,35,36)(20,37,31,21)(23,34,40,39)(25,41,42,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390256);