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×#29840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2154 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);(3,4,5,6)(7,8,10,11)(9,12,13,14)(15,16)(17,21)(18,27)(19,22)(20,23)(24,28)(25,29)(26,30);(8,11)(12,14)(15,17,20,26)(16,21,23,30)(18,25,19,28)(22,24,27,29);(4,6)(9,13)(12,14)(15,16)(17,24)(18,22)(19,27)(20,23)(21,28)(25,30)(26,29);(7,9,10,13)(8,12,11,14)(15,18,20,19)(16,22,23,27)(17,25,26,28)(21,24,30,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342679);