Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2304 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)(14,26)(15,30)(16,20)(17,21)(19,28)(22,38)(23,33)(24,40)(25,34)(27,32)(29,36)(31,37)(35,39)(41,42);(3,5,6,10)(4,7,8,9)(11,13,16,25,17,23,15,24)(12,18,20,34,21,33,30,40)(14,27,28,39,29,41,31,22)(19,35,36,42,37,38,26,32);(11,14,15,28,17,29,16,31)(12,19,20,26,21,37,30,36)(13,22,24,27,23,39,25,41)(18,32,34,38,33,42,40,35);(4,8)(7,9)(13,23)(18,33)(22,39)(24,25)(27,41)(32,42)(34,40)(35,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366791);