Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18518 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3144 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)(5,9)(8,10)(11,12)(13,19)(14,18)(15,20)(16,21)(17,22)(23,26)(24,25);(3,4,6,8)(5,10,9,7)(11,13,17,24)(12,18,22,26)(14,15,23,16)(19,21,25,20);(11,14)(12,19)(13,15)(16,24)(17,23)(18,21)(20,26)(22,25);(11,15,17,16)(12,20,22,21)(13,23,24,14)(18,19,26,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255257);