Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29837 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,5,6)(7,8,10,11)(9,12,13,14)(15,16)(17,20)(18,24)(19,25)(21,23)(22,28)(26,29)(27,30);(8,11)(12,14)(15,17,20,16)(18,26,28,30)(19,25,21,23)(22,29,24,27);(4,6)(9,13)(12,14)(15,18,20,28)(16,22,17,24)(19,29,21,27)(23,26,25,30);(7,9,10,13)(8,12,11,14)(15,19,20,21)(16,23,17,25)(18,27,28,29)(22,30,24,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342674);