Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2341 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);(3,4,8,14,9,13,23,30)(5,11,16,28,6,17,19,31)(7,12,21,29,20,25,22,32)(10,24,26,18,27,33,34,15);(3,5,9,6)(4,10,13,27)(7,15,20,18)(8,19,23,16)(11,25,17,12)(14,26,30,34)(21,33,22,24)(28,32,31,29);(3,6,9,5)(4,11,13,17)(7,15,20,18)(8,16,23,19)(10,25,27,12)(14,31,30,28)(21,33,22,24)(26,29,34,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400795);