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×#40815 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5060 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);(12,15)(16,18)(19,20,25,28)(21,27,22,26)(23,33,34,24)(29,32,31,30);(3,4)(5,9)(6,7)(8,10)(19,21,25,22)(20,26,28,27)(23,29,34,31)(24,32,33,30);(4,7)(8,9)(11,12,14,15)(13,16,17,18)(19,22,25,21)(20,27,28,26)(23,29,34,31)(24,32,33,30);(3,4)(5,8)(6,7)(9,10)(11,13,14,17)(12,16,15,18)(19,23)(20,24)(21,29)(22,31)(25,34)(26,32)(27,30)(28,33);(3,5)(4,8)(6,10)(7,9)(19,24)(20,23)(21,30)(22,32)(25,33)(26,31)(27,29)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546225);