Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(5,10)(8,9)(11,12)(13,17)(14,21)(15,23)(16,19)(18,20)(22,25)(24,26);(3,4,6,9)(5,8,10,7)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21);(11,14,16,22)(12,18,19,26)(13,20,15,24)(17,25,23,21);(4,8)(5,10)(7,9)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265412);