Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(8,9)(12,13)(15,16)(17,25)(18,21)(19,23)(20,29)(22,27)(24,28)(26,30);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17,19,18)(16,21,23,25)(20,26,28,27)(22,24,30,29);(4,6)(7,9,11,8)(10,12,14,13)(15,18,19,17)(16,22,23,30)(20,26,28,27)(21,24,25,29);(7,10,11,14)(8,12,9,13)(17,18)(21,25)(22,30)(26,27)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342624);