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×#41426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(13,17)(15,18)(19,20,24,25)(21,26,29,33)(22,30,32,23)(27,28,31,34);(4,7)(8,9)(11,12,14,16)(13,15,17,18)(20,25)(23,30)(26,33)(28,34);(3,4)(5,8)(6,7)(9,10)(11,13,14,17)(12,15,16,18)(19,21)(20,26)(22,31)(23,28)(24,29)(25,33)(27,32)(30,34);(3,5)(4,8)(6,10)(7,9)(19,22)(20,23)(21,27)(24,32)(25,30)(26,34)(28,33)(29,31);(19,23)(20,22)(21,28)(24,30)(25,32)(26,31)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546828);