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×#50888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7946 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)(9,13)(10,11)(12,14)(15,16,20,21)(17,22,25,29)(18,23,24,19)(26,27,28,30);(8,11)(12,13)(16,21)(17,25)(19,23)(26,28);(3,4,5,6)(15,17)(16,22)(18,28)(19,27)(20,25)(21,29)(23,30)(24,26);(7,8)(9,12)(10,11)(13,14)(15,18,20,24)(16,23,21,19)(17,26,25,28)(22,27,29,30);(7,9,10,14)(8,12,11,13)(15,19,20,23)(16,24,21,18)(17,27,25,30)(22,26,29,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613854);