Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2020 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#121 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4);(5,6)(7,8)(9,10,15,20)(11,21,25,32)(12,26,30,35)(13,19,14,18)(16,24,33,23)(17,29,36,28)(22,37,39,31)(27,34,38,40);(6,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,32)(19,21)(20,33)(22,29)(26,37)(28,39)(30,38)(31,35)(36,40);(9,12,15,30)(10,17,20,36)(11,22,25,39)(13,29,14,28)(16,31,33,37)(18,35,19,26)(21,34,32,40)(23,27,24,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266214);