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×#50897 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8904 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)(10,14)(11,12)(15,16,20,18)(17,23,25,21)(19,26,27,22)(24,28,30,29);(7,9)(8,10)(11,14)(12,13)(16,18)(21,23)(22,26)(28,29);(3,4,5,6)(8,12)(9,14)(15,17)(16,21)(18,23)(19,24)(20,25)(22,28)(26,29)(27,30);(7,9)(8,13)(10,12)(11,14)(15,18,20,16)(17,23,25,21)(19,26,27,22)(24,29,30,28);(7,10,11,13)(8,9,12,14)(15,19,20,27)(16,22,18,26)(17,24,25,30)(21,28,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613910);