Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19125 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2122 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)(5,8)(9,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(3,4,8,10,6,9,5,7);(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260788);