Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13732 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 74 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,14,18)(13,23,25,46)(15,32,29,59)(16,35,30,60)(17,36,31,22)(19,40,37,26)(20,43,38,24)(21,45,39,33)(27,57,53,74)(28,58,54,49)(34,63,51,62)(41,69,65,73)(42,56,64,68)(44,52,61,70)(47,67,71,50)(48,66,72,55);(3,5,6,10)(4,7,8,9)(12,18)(19,37)(20,38)(21,39)(22,36)(23,46)(32,59)(35,60)(41,65)(42,64)(44,61)(47,71)(48,72)(49,58)(57,74)(62,63);(11,13,24,50,16,27,51,68,31,54,70,69,45,66,29,40,14,25,43,67,30,53,34,56,17,28,52,73,33,55,15,26)(12,19,32,48,21,41,61,58,36,64,63,74,60,71,38,46,18,37,59,72,39,65,44,49,22,42,62,57,35,47,20,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420254);