Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#22281 |
| 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 | C2×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2406 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)(6,9)(7,10)(11,12,16,20,17,21,15,30)(13,18,24,32,25,33,23,39)(14,26,29,34,27,40,28,19)(22,35,38,41,36,42,37,31);(3,5,8,4)(6,9,7,10)(11,13,16,24,17,25,15,23)(12,18,20,32,21,33,30,39)(14,22,29,38,27,36,28,37)(19,31,26,35,34,41,40,42);(3,6,8,7)(4,9,5,10)(11,14,15,28,17,27,16,29)(12,19,20,26,21,34,30,40)(13,22,23,37,25,36,24,38)(18,31,32,35,33,41,39,42);(3,7,8,6)(4,9,5,10)(14,27)(19,34)(22,36)(26,40)(28,29)(31,41)(35,42)(37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283652);