Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,17)(15,18)(19,20,24,28)(21,25,31,38)(22,26,33,40)(23,34,35,27)(29,41,43,36)(30,44,45,37)(32,46,47,39)(42,48,50,49);(3,5,8,15)(4,9,11,17)(6,14,7,13)(10,18,16,12)(19,21)(20,25)(22,29)(23,30)(24,31)(26,36)(27,37)(28,38)(32,42)(33,43)(34,44)(35,45)(39,48)(40,41)(46,49)(47,50);(19,22,24,33)(20,26,28,40)(21,29,31,43)(23,32,35,47)(25,36,38,41)(27,39,34,46)(30,42,45,50)(37,48,44,49);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(19,23,24,35)(20,27,28,34)(21,30,31,45)(22,32,33,47)(25,37,38,44)(26,39,40,46)(29,42,43,50)(36,48,41,49)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254134);