Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13731 |
| 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 | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(12,17)(13,18)(14,22)(15,25)(19,34)(20,37)(21,30)(23,39)(24,33)(26,38)(27,28)(29,35)(31,32)(36,42)(40,41);(3,5,6,10)(4,7,8,9);(11,12,18,28,14,19,31,37,25,35,39,42,38,40,24,30,16,21,33,41,26,36,23,29,15,20,32,34,22,27,13,17)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420253);