Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11594 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,22)(14,19)(15,28)(16,32)(17,21)(18,24)(20,30)(23,36)(25,42)(26,39)(27,37)(29,35)(31,38)(33,40)(34,41);(11,13,17,26)(12,18,21,34)(14,25,16,23)(15,29,31,40)(19,35,32,33)(20,36,37,42)(22,38,39,28)(24,27,41,30);(11,14,17,16)(12,19,21,32)(13,23,26,25)(15,27,31,30)(18,33,34,35)(20,28,37,38)(22,36,39,42)(24,40,41,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388250);