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×#48190 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(12,15)(16,18)(19,20,24,29)(21,26,31,25)(22,27,32,33)(23,30,34,28);(3,4)(5,9)(6,10)(7,8)(19,21,24,31)(20,25,29,26)(22,30,32,28)(23,33,34,27);(4,8)(5,10)(11,12,14,15)(13,16,17,18)(19,21)(20,26)(22,28)(23,27)(24,31)(25,29)(30,32)(33,34);(3,5)(4,6)(7,10)(8,9)(11,13,14,17)(12,16,15,18)(19,22)(20,27)(21,30)(23,25)(24,32)(26,34)(28,31)(29,33);(3,6)(4,5)(7,9)(8,10)(19,23,24,34)(20,28,29,30)(21,27,31,33)(22,25,32,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610964);