Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42733 |
| 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 | |
4686 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);(3,4,7,5)(6,8,10,9)(13,17)(16,18)(19,20)(21,29)(22,31)(23,28)(24,25)(26,32)(27,33)(30,34);(3,4,7,5)(6,9,10,8)(11,12,15,14)(13,16,17,18)(20,25)(23,34)(26,29)(27,31);(4,5)(8,9)(11,13)(12,16)(14,18)(15,17)(19,21,24,32)(20,26,25,29)(22,28,33,30)(23,31,34,27);(3,5,7,4)(6,9,10,8)(11,14,15,12)(13,18,17,16)(19,22,24,33)(20,27,25,31)(21,30,32,28)(23,26,34,29);(3,6,7,10)(4,8,5,9)(19,23,24,34)(20,28,25,30)(21,31,32,27)(22,26,33,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548433);