Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#40516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7356 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);(3,4,5,6)(15,16)(17,21)(18,25)(19,20)(22,27)(23,30)(24,28)(26,29);(7,8)(9,13)(10,11)(12,14)(16,20)(21,28)(22,25)(29,30);(3,4)(5,6)(8,11)(12,13)(15,17,19,24)(16,21,20,28)(18,26,27,23)(22,30,25,29);(7,8)(9,12)(10,11)(13,14)(15,18,19,27)(16,22,20,25)(17,23,24,26)(21,29,28,30);(7,9)(8,12)(10,14)(11,13)(15,16)(17,21)(18,22)(19,20)(23,29)(24,28)(25,27)(26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546128);