Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7792 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);(4,5)(9,13)(12,14);(3,4,6,5)(7,8,11,10)(9,12,13,14)(16,20)(21,27)(22,28)(29,30);(3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,16)(17,21)(18,22)(19,20)(23,29)(24,27)(25,30)(26,28);(7,10,11,8)(9,14,13,12)(15,17,19,24)(16,21,20,27)(18,25,26,23)(22,30,28,29);(15,18,19,26)(16,22,20,28)(17,23,24,25)(21,29,27,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536684);