Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7032 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)(15,16,20,25)(17,22,28,21)(18,23,24,19)(26,27,29,30);(7,8)(9,10)(11,12)(13,14)(15,17,20,28)(16,21,25,22)(18,29,24,26)(19,30,23,27);(3,4,5,6)(8,12)(9,13)(15,17)(16,22)(18,29)(19,27)(20,28)(21,25)(23,30)(24,26);(7,9,11,13)(8,10,12,14)(15,18,20,24)(16,23,25,19)(17,26,28,29)(21,30,22,27);(7,10)(8,13)(9,12)(11,14)(15,19,20,23)(16,24,25,18)(17,27,28,30)(21,29,22,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608839);