Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#39873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7438 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)(15,16)(17,23)(18,24)(19,22)(20,21)(25,29)(26,30)(27,28);(8,12)(10,13)(16,21)(17,24)(22,28)(25,30);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,18)(19,25)(20,24)(21,23)(22,26)(27,30)(28,29);(7,9,11,14)(8,13,12,10)(15,18)(16,17)(19,26)(20,23)(21,24)(22,25)(27,29)(28,30);(7,10,11,13)(8,14,12,9)(15,19,20,27)(16,22,21,28)(17,25,24,30)(18,26,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544815);