Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#33468 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C2×#442 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,16,19,26)(6,20)(7,13)(8,12)(9,24)(10,15)(11,17,28,34)(14,22)(18,31,25,32)(21,30)(23,27,33,29);(3,5)(4,11)(6,17)(7,18)(8,23)(9,25)(10,19)(12,26)(13,27)(14,31)(15,28)(16,30)(20,32)(21,33)(22,34)(24,29);(3,6,9,8,10,22,7,21)(4,12,13,20,15,30,24,14)(5,17,25,23,19,34,18,33)(11,26,27,32,28,16,29,31);(3,7,10,9)(4,13,15,24)(5,18,19,25)(6,21,22,8)(11,27,28,29)(12,20,30,14)(16,31,26,32)(17,33,34,23)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529827);