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×#42980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7318 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)(7,8,11,13)(9,10,14,12)(15,16)(17,25)(18,27)(19,24)(20,21)(22,28)(23,29)(26,30);(7,9,11,14)(8,12,13,10)(16,21)(19,30)(22,25)(23,27);(3,4,5,6)(7,9)(8,10)(11,14)(12,13)(15,17,20,28)(16,22,21,25)(18,24,29,26)(19,27,30,23);(7,10)(8,9)(11,12)(13,14)(15,18,20,29)(16,23,21,27)(17,26,28,24)(19,22,30,25);(15,19,20,30)(16,24,21,26)(17,27,28,23)(18,22,29,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548896);