Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#52064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#560 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8)(9,10)(11,12)(13,14)(15,16,20,21)(17,18,22,26)(19,27,28,23)(24,30,29,25);(8,12)(9,13)(16,21)(17,22)(19,28)(25,30);(3,4,5,6)(15,17)(16,18)(19,24)(20,22)(21,26)(23,25)(27,30)(28,29);(7,9,11,13)(8,10,12,14)(15,18)(16,22)(17,21)(19,25)(20,26)(23,29)(24,27)(28,30);(7,10)(8,13)(9,12)(11,14)(15,19,20,28)(16,23,21,27)(17,24,22,29)(18,25,26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615319);