Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,25,31,34)(22,26,23,27)(29,32,30,33);(3,5,7,6)(4,8,10,9)(11,13,15,14)(12,16,18,17)(19,21,24,31)(20,25,28,34)(22,30,23,29)(26,33,27,32);(3,5,7,6)(4,9,10,8)(11,14,15,13)(12,17,18,16)(21,29)(22,23)(25,32)(26,27)(30,31)(33,34);(3,6,7,5)(4,9,10,8)(12,18)(16,17)(19,22,24,23)(20,26,28,27)(21,30,31,29)(25,33,34,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354026);