Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11319 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #401 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1443 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,11,20,27)(5,9,15,23,14,22,19,28)(6,16,18,31,17,10,24,26)(13,21,30,33,29,25,32,34);(3,5)(4,9,11,22)(6,13,17,29)(7,19)(8,14)(10,25)(12,28,27,23)(15,20)(16,21)(18,32,24,30)(26,33)(31,34);(3,6)(4,10,11,16)(5,13,14,29)(7,18)(8,17)(9,21)(12,26,27,31)(15,30,19,32)(20,24)(22,25)(23,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265563);