Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#12847 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1198 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)(5,8)(6,7)(9,10)(11,12,14,19,15,20,25,32,16,18,24,30,26,31,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42);(4,7)(8,10)(11,13)(12,17)(14,22)(15,23)(16,21)(18,27)(19,28)(20,29)(24,33)(25,35)(26,34)(30,37)(31,38)(32,39)(36,41)(40,42);(3,5,6,9)(4,8,7,10)(12,18)(13,21)(14,24)(20,31)(23,34)(25,36)(28,37)(39,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401034);