Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7481 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7484 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | #532 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,6,11,8,14,23,30)(5,15,17,9,20,25,33,27)(7,13,21,10,24,12,22,28)(16,31,19,29,34,26,18,32);(3,5,7,18,8,20,24,19)(4,9,12,26,14,27,13,31)(6,16,21,33,23,34,22,17)(10,15,11,29,28,25,30,32)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58983);