Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(11,12,16,20,15,19,29,36)(13,21,24,38,23,17,31,33)(14,25,28,37,27,18,30,35)(22,32,39,41,26,34,40,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,26)(15,23)(16,24)(18,34)(19,21)(20,33)(22,27)(25,32)(28,40)(29,31)(30,39)(35,42)(36,38)(37,41);(4,7)(8,10)(11,14,15,27)(12,18,19,25)(13,22,23,26)(16,30,29,28)(17,32,21,34)(20,37,36,35)(24,40,31,39)(33,42,38,41);(3,5,6,9)(4,8,7,10)(12,19)(13,23)(18,25)(20,36)(22,26)(24,31)(35,37)(39,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281686);