Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C8 |
| direct product | C2×#19724 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,13)(6,10)(7,11)(8,18)(9,12)(14,16)(15,17)(19,20)(21,27)(22,32)(23,28)(24,33)(25,29)(26,30)(31,34);(3,5,8,10,9,15,7,14)(4,6,11,13,12,16,18,17)(19,21,24,30,25,31,23,22)(20,26,28,27,29,32,33,34);(3,6,9,16)(4,10,12,14)(5,11,15,18)(7,17,8,13)(19,22,23,31,25,30,24,21)(20,27,33,26,29,34,28,32);(3,7,9,8)(4,11,12,18)(5,14,15,10)(6,13,16,17)(19,23,25,24)(20,28,29,33)(21,30,31,22)(26,34,32,27)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264556);