Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17897 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,7,5,6,8,10,9)(11,12)(13,21)(14,18)(15,27)(16,19)(17,23)(20,29)(22,35)(24,34)(25,33)(26,39)(28,31)(30,37)(32,38)(36,42)(40,41);(4,8)(5,9)(11,13,16,24)(12,17,19,31)(14,22,25,36)(15,28,29,23)(18,30,33,41)(20,34,27,21)(26,40,38,37)(32,42,39,35);(3,5,7,8,6,9,10,4)(11,14,16,25)(12,18,19,33)(13,22,24,36)(15,26,29,38)(17,30,31,41)(20,32,27,39)(21,35,34,42)(23,37,28,40);(12,19)(14,25)(17,31)(20,27)(21,34)(22,36)(26,38)(37,40)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251787);