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×#30626 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3060 | 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)(13,18)(14,27)(15,19)(16,21)(17,22)(20,28)(23,32)(24,38)(25,33)(26,34)(29,35)(30,36)(31,37)(39,41)(40,42);(3,5)(4,8)(6,9)(7,10)(11,13,17,26)(12,18,22,34)(14,24,29,40)(15,25,16,23)(19,33,21,32)(20,31,36,41)(27,38,35,42)(28,37,30,39);(3,6,7,8)(4,5,9,10)(12,19)(13,23)(15,16)(20,27)(21,22)(24,37)(25,26)(28,30)(32,33)(35,36)(38,42)(39,40);(4,9)(5,10)(11,14,17,29)(12,20,22,36)(13,24,26,40)(15,30,16,28)(18,31,34,41)(19,27,21,35)(23,37,25,39)(32,42,33,38)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408534);