Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7227 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C22 | #1289 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2);(3,4)(5,15)(6,20,22,46)(7,25)(8,13)(9,14)(10,16,28,40)(11,24,31,48)(12,33)(17,43)(18,37)(19,38)(21,34)(23,50,49,32)(26,35)(27,53)(29,56,54,64)(30,42,55,61)(36,57)(39,59)(41,63,62,52)(44,60)(45,65)(47,51,66,58);(3,5,7,17,8,18,26,44,9,19,27,45,21,39,33,57)(4,10,12,29,34,55,53,63,14,28,35,54,13,30,25,52)(6,16,23,41,24,42,51,64,22,40,49,62,48,61,58,56)(11,15,32,36,46,59,66,65,31,38,50,60,20,37,47,43)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34537);