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×#41039 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5820 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);(11,12)(13,15)(14,16)(17,18)(19,20,25,30)(21,27,22,26)(23,24,29,28)(31,32,33,34);(3,4)(5,9)(6,7)(8,10)(11,12)(13,17)(14,16)(15,18)(19,21,25,22)(20,26,30,27)(23,31,29,33)(24,34,28,32);(4,7)(8,9)(11,12,14,16)(13,15,18,17)(19,22,25,21)(20,27,30,26)(23,31,29,33)(24,34,28,32);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23,25,29)(20,28,30,24)(21,31,22,33)(26,32,27,34);(3,5)(4,8)(6,10)(7,9)(11,13,14,18)(12,15,16,17)(19,24,25,28)(20,29,30,23)(21,32,22,34)(26,31,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546230);