Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#21232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5058 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);(4,6)(7,8,9,10)(11,12)(13,23)(14,28)(15,20)(16,21)(17,22)(18,26)(19,30)(24,32)(25,34)(27,33)(29,36)(31,35)(37,38)(39,41)(40,42);(3,4,5,6)(8,10)(11,13,15,26,17,27,16,25)(12,18,21,23,22,34,20,33)(14,24,29,39,31,40,30,38)(19,32,35,37,36,42,28,41);(11,14,15,29,17,31,16,30)(12,19,21,35,22,36,20,28)(13,24,26,39,27,40,25,38)(18,32,23,37,34,42,33,41);(11,15,17,16)(12,20,22,21)(13,25,27,26)(14,29,31,30)(18,23,34,33)(19,28,36,35)(24,38,40,39)(32,37,42,41)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279190);