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×#37731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7384 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);(8,11)(12,14)(15,16,20,23)(17,18,25,21)(19,22,27,29)(24,26,30,28);(4,6)(15,17,20,25)(16,21,23,18)(19,24,27,30)(22,28,29,26);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,24)(20,25)(21,23)(22,26)(27,30)(28,29);(7,9,10,13)(8,12,11,14)(15,18)(16,17)(19,26)(20,21)(22,24)(23,25)(27,28)(29,30);(3,4)(5,6)(15,19,20,27)(16,22,23,29)(17,24,25,30)(18,26,21,28)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542293);