Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#9973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8,12,9)(10,13,11,14)(15,16,21,18)(17,24,27,30)(19,29,20,23)(22,26,28,25);(8,13)(9,14)(10,11)(15,17,21,27)(16,22,18,28)(19,26,20,25)(23,30,29,24);(4,6)(7,9,12,8)(10,14,11,13)(15,18,21,16)(17,24,27,30)(19,29,20,23)(22,25,28,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139401);