Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51265 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(13,17)(16,18)(19,20,24,28)(21,22,26,25)(23,27,31,34)(29,30,33,32);(3,4,7,10)(5,8,6,9)(11,12,15,14)(13,16,17,18)(21,26)(22,25)(23,31)(27,34);(3,5,7,6)(4,8,10,9)(11,13)(12,16)(14,18)(15,17)(19,21)(20,25)(22,28)(23,29)(24,26)(27,32)(30,34)(31,33);(3,4)(5,8)(6,9)(7,10)(19,22,24,25)(20,26,28,21)(23,30,31,32)(27,33,34,29);(3,6,7,5)(4,9,10,8)(11,14,15,12)(13,18,17,16)(19,23,24,31)(20,27,28,34)(21,29,26,33)(22,30,25,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614016);