Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | #532 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
681 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);(3,4,6,12)(5,15,17,29)(7,23,21,31)(8,13,25,11)(9,14,22,30)(10,19,26,33)(16,28,24,32)(18,34,20,27);(3,5,7,18,9,19,8,24)(4,10,13,27,14,29,23,28)(6,20,21,33,22,16,25,17)(11,26,30,34,31,15,12,32)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59011);