Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12)(13,18)(14,24)(15,28)(16,32)(17,21)(19,26)(20,30)(22,38)(23,37)(25,36)(27,40)(29,33)(31,42)(34,41)(35,39);(3,5)(4,8)(6,9)(7,10)(11,13,17,16)(12,18,21,32)(14,25,27,22)(15,23,31,30)(19,34,35,33)(20,28,37,42)(24,36,40,38)(26,41,39,29);(3,6,7,4)(5,9,10,8)(11,14)(12,19)(13,22)(15,29)(16,25)(17,27)(18,33)(20,36)(21,35)(23,39)(24,28)(26,30)(31,41)(32,34)(37,38)(40,42);(5,10)(8,9)(11,13,17,16)(12,18,21,32)(14,22,27,25)(15,23,31,30)(19,33,35,34)(20,28,37,42)(24,38,40,36)(26,29,39,41)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530996);