Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23814 |
| 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 | |
2778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 46 points:
(1,2);(7,8,10,11)(9,12,13,14)(15,16,19,23,20,24,34,40)(17,25,28,36,27,38,32,21)(18,29,30,37,31,39,33,22)(26,35,43,46,42,45,44,41);(3,4)(5,6)(8,11)(12,14)(15,17,20,27)(16,21,24,36)(18,26,31,42)(19,32,34,28)(22,35,37,45)(23,38,40,25)(29,41,39,46)(30,44,33,43);(4,6)(15,18,20,31)(16,22,24,37)(17,26,27,42)(19,33,34,30)(21,35,36,45)(23,39,40,29)(25,41,38,46)(28,44,32,43);(7,9,10,13)(8,12,11,14)(17,27)(21,36)(25,38)(26,42)(28,32)(35,45)(41,46)(43,44)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339800);