Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#44579 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7428 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)(8,11)(12,14)(15,16,20,24)(17,21,25,29)(18,22,27,30)(19,26,28,23);(3,4,5,6)(17,25)(19,28)(21,29)(23,26);(7,8)(9,12)(10,11)(13,14)(15,17,20,25)(16,21,24,29)(18,26,27,23)(19,22,28,30);(3,4)(5,6)(15,18)(16,22)(17,23)(19,21)(20,27)(24,30)(25,26)(28,29);(7,9,10,13)(8,12,11,14)(15,19)(16,23)(17,22)(18,21)(20,28)(24,26)(25,30)(27,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7575974);