Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37522 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4886 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);(5,9)(8,10)(11,12,14,16)(13,15,17,18)(19,20)(21,28)(22,26)(23,33)(24,34)(25,27)(29,30)(31,32);(3,4)(5,8)(6,7)(9,10)(19,21,25,22)(20,26,27,28)(23,29,34,31)(24,30,33,32);(4,7)(8,10)(11,12)(13,15)(14,16)(17,18)(19,22,25,21)(20,26,27,28)(23,29,34,31)(24,32,33,30);(3,4)(5,8)(6,7)(9,10)(19,23)(20,24)(21,29)(22,31)(25,34)(26,30)(27,33)(28,32);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,15,16,18)(19,24)(20,23)(21,30)(22,32)(25,33)(26,29)(27,34)(28,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540887);