Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#21043 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2342 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)(5,8)(6,7)(9,10)(11,12,16,21,17,20,15,29)(13,18,25,32,23,33,24,39)(14,26,27,19,28,35,30,34)(22,36,38,31,37,41,40,42);(4,7)(8,10)(11,13,16,25,17,23,15,24)(12,18,21,32,20,33,29,39)(14,22,27,38,28,37,30,40)(19,31,35,41,34,42,26,36);(3,5,6,9)(4,8,7,10)(11,14)(12,19)(13,22)(15,30)(16,27)(17,28)(18,31)(20,34)(21,35)(23,37)(24,40)(25,38)(26,29)(32,41)(33,42)(36,39);(12,20)(13,23)(19,34)(21,29)(22,37)(24,25)(26,35)(38,40)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278804);