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×#36045 |
| 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 | |
7814 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)(7,8,11,14)(9,12,10,13)(15,16)(17,22)(18,23)(19,24)(20,21)(25,27)(26,30)(28,29);(3,4,6,5)(7,9,11,10)(8,12,14,13)(16,21)(17,23)(24,29)(25,30);(3,5,6,4)(7,8)(9,12)(10,13)(11,14)(15,17,20,23)(16,22,21,18)(19,25,28,30)(24,27,29,26);(7,10,11,9)(8,13,14,12)(15,18,20,22)(16,23,21,17)(19,27,28,26)(24,25,29,30);(15,19)(16,24)(17,25)(18,26)(20,28)(21,29)(22,27)(23,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536678);