Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1664 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,26,24)(23,29,32,36)(25,33,35,28)(27,34,30,31);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,23,26,32)(22,27,24,30)(25,34,35,31)(28,36,33,29);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(21,24,26,22)(23,30,32,27)(25,28,35,33)(29,34,36,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116343);