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×#46149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8,11,12)(9,10,14,13)(15,16)(17,18)(19,27)(20,21)(22,25)(23,28)(24,29)(26,30);(8,12)(10,13)(16,21)(18,22)(23,27)(26,29);(3,4,5,6)(7,8,11,12)(9,13,14,10)(15,17)(16,22)(18,21)(19,24)(20,25)(23,29)(26,27)(28,30);(7,9)(8,13)(10,12)(11,14)(15,18,20,22)(16,17,21,25)(19,26,28,29)(23,24,27,30);(7,10,11,13)(8,9,12,14)(15,19,20,28)(16,23,21,27)(17,24,25,30)(18,26,22,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610005);