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×#50361 |
| 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);(3,4,7,9)(5,8,10,6)(12,15)(16,18)(19,20)(21,28)(22,27)(23,26)(24,25)(29,34)(30,33)(31,32);(5,10)(6,8)(19,21,25,23)(20,26,24,28)(22,30,32,29)(27,34,31,33);(3,5,7,10)(4,6,9,8)(11,12)(13,16)(14,15)(17,18)(19,22,25,32)(20,27,24,31)(21,29,23,30)(26,33,28,34);(3,5)(4,8)(6,9)(7,10)(19,23,25,21)(20,26,24,28)(22,29,32,30)(27,34,31,33);(3,6,7,8)(4,5,9,10)(11,13,14,17)(12,16,15,18)(19,24,25,20)(21,26,23,28)(22,31,32,27)(29,33,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613365);