Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(8,10)(12,13)(15,16,20,18)(17,21,23,25)(19,26,27,22)(24,30,29,28);(7,8,11,10)(9,13,14,12)(16,18)(17,23)(19,27)(28,30);(3,4,5,6)(15,17)(16,21)(18,25)(19,24)(20,23)(22,28)(26,30)(27,29);(7,9,11,14)(8,12,10,13)(15,18,20,16)(17,21,23,25)(19,26,27,22)(24,28,29,30);(7,10,11,8)(9,13,14,12)(15,19,20,27)(16,22,18,26)(17,24,23,29)(21,28,25,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608063);