Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7502 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);(4,6)(9,13)(12,14)(15,16,19,22)(17,23,25,20)(18,26,28,21)(24,29,27,30);(8,11)(9,13)(15,17,19,25)(16,20,22,23)(18,27,28,24)(21,30,26,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,20)(18,24)(19,25)(21,29)(22,23)(26,30)(27,28);(7,9,10,13)(8,12,11,14)(15,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,28)(27,30);(7,8)(9,12)(10,11)(13,14)(15,18,19,28)(16,21,22,26)(17,24,25,27)(20,29,23,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609878);