Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37641 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7110 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)(9,13)(12,14)(16,19)(18,26)(21,24)(23,27);(7,8)(9,12)(10,11)(13,14)(15,16,20,19)(17,21,25,24)(18,22,26,28)(23,29,27,30);(3,4)(5,6)(8,11)(12,14)(15,17,20,25)(16,21,19,24)(18,27,26,23)(22,30,28,29);(7,8)(9,12)(10,11)(13,14)(15,18)(16,22)(17,23)(19,28)(20,26)(21,29)(24,30)(25,27);(7,9,10,13)(8,12,11,14)(15,19,20,16)(17,24,25,21)(18,28,26,22)(23,30,27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541133);