Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(4,6)(9,13)(12,14)(15,16)(17,21)(18,19)(20,22);(8,10)(9,13)(16,18)(20,21);(3,4,5,6)(7,8,11,10)(9,12,13,14)(15,16,19,18)(17,20,22,21);(7,9,11,13)(8,12,10,14)(15,17)(16,20)(18,21)(19,22);(7,10,11,8)(9,14,13,12)(15,18,19,16)(17,21,22,20)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610225);