Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,17,19,16,22,14,20)(13,18,26,34,27,35,24,32)(15,21,30,36,29,39,28,37)(23,38,40,31,41,42,25,33);(3,5,7,6)(4,8,9,10)(11,13)(12,18)(14,24)(15,23)(16,27)(17,26)(19,34)(20,32)(21,38)(22,35)(25,28)(29,41)(30,40)(31,36)(33,37)(39,42);(3,6,7,5)(4,8,9,10)(12,19)(13,23)(14,17)(15,28)(18,31)(20,22)(24,40)(25,26)(27,41)(29,30)(32,42)(33,35)(34,38)(36,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406608);