Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13368 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
970 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);(3,4,6,8)(5,9,10,7)(11,12)(13,21)(14,25)(15,30)(16,19)(17,22)(18,29)(20,28)(23,41)(24,38)(26,37)(27,36)(31,35)(32,39)(33,42)(34,40);(3,5,6,10)(4,7,8,9)(14,26)(17,32)(18,34)(22,39)(25,37)(27,42)(29,40)(33,36);(11,13,16,24)(12,17,19,32)(14,27,26,42)(15,31,28,23)(18,35,34,41)(20,36,30,33)(21,37,38,25)(22,40,39,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406154);