Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3894 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);(3,4)(5,9)(6,8)(7,10)(11,12)(13,18)(14,25)(15,20)(16,29)(17,21)(19,27)(22,38)(23,32)(24,34)(26,36)(28,37)(30,33)(31,39)(35,41)(40,42);(3,5,6,10)(4,7,8,9)(11,13)(12,18)(14,22)(15,23)(16,30)(17,24)(19,35)(20,32)(21,34)(25,38)(26,39)(27,41)(28,40)(29,33)(31,36)(37,42);(11,14,17,28)(12,19,21,36)(13,22,24,40)(15,27,16,26)(18,31,34,35)(20,37,29,25)(23,41,30,39)(32,38,33,42);(4,8)(7,9)(11,15,17,16)(12,20,21,29)(13,23,24,30)(14,26,28,27)(18,32,34,33)(19,25,36,37)(22,39,40,41)(31,42,35,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358940);