Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#17193 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4698 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);(4,6)(8,11)(9,13)(15,16,18,20)(17,21,22,19)(23,24,26,27)(25,28,29,30);(8,11)(9,13)(17,22)(19,21)(24,27)(25,29);(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,25)(24,28)(26,29)(27,30);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,17,18,22)(16,19,20,21)(23,24)(25,30)(26,27)(28,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752172);