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×#51201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7378 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)(9,13)(10,12)(11,14)(15,16,19,20)(17,21,23,27)(18,22,26,28)(24,29,25,30);(7,8,10,12)(9,11,14,13)(16,20)(17,23)(22,28)(24,25);(3,4,5,6)(15,17)(16,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29);(7,8)(9,11)(10,12)(13,14)(15,18,19,26)(16,22,20,28)(17,24,23,25)(21,29,27,30);(7,9)(8,11)(10,14)(12,13)(15,16)(17,21)(18,22)(19,20)(23,27)(24,29)(25,30)(26,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614005);