Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30623 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,6)(7,9)(8,10)(11,12)(13,23)(14,28)(15,21)(16,19)(17,22)(18,26)(20,32)(24,35)(25,40)(27,34)(29,42)(30,36)(31,38)(33,39)(37,41);(3,5)(4,8)(6,9)(7,10)(11,13)(12,18)(14,29)(15,26)(16,27)(17,24)(19,35)(20,37)(21,23)(22,34)(25,30)(28,40)(31,41)(32,39)(33,38)(36,42);(3,6,7,8)(4,5,9,10)(12,19)(13,24)(16,17)(20,36)(21,22)(25,39)(26,27)(28,38)(29,41)(31,32)(34,35)(40,42);(4,9)(5,10)(11,14)(12,20)(13,25)(15,30)(16,32)(17,31)(18,33)(19,36)(21,38)(22,28)(23,37)(24,39)(26,29)(27,41)(34,42)(35,40)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408537);