Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#28170 |
| 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 | |
2398 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)(9,13)(10,11)(12,14)(15,16,20,24)(17,21,26,35)(18,28,30,38)(19,31,33,23)(22,32,37,29)(25,39,40,46)(27,43,42,36)(34,44,45,41);(3,4,5,6)(8,11)(12,13)(15,17)(16,21)(18,25)(19,27)(20,26)(22,34)(23,36)(24,35)(28,39)(29,41)(30,40)(31,43)(32,44)(33,42)(37,45)(38,46);(4,6)(15,18,20,30)(16,22,24,37)(17,25,26,40)(19,32,33,29)(21,34,35,45)(23,38,31,28)(27,44,42,41)(36,46,43,39);(7,9,10,14)(8,12,11,13)(17,26)(21,35)(25,40)(27,42)(34,45)(36,43)(39,46)(41,44)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370039);