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×#50090 |
| 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 | |
5478 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);(3,4)(5,7)(6,10)(8,9)(19,20,24,27)(21,23,29,25)(22,26,32,34)(28,31,30,33);(3,5)(4,7)(6,8)(9,10)(12,15)(13,17)(19,21,24,29)(20,25,27,23)(22,28,32,30)(26,33,34,31);(4,9)(5,6)(11,12,14,15)(13,16,17,18)(19,21)(20,25)(22,30)(23,27)(24,29)(26,31)(28,32)(33,34);(3,6,8,5)(4,10,9,7)(11,12)(13,16)(14,15)(17,18)(19,22)(20,26)(21,28)(23,31)(24,32)(25,33)(27,34)(29,30);(3,7,8,10)(4,5,9,6)(11,13)(12,16)(14,17)(15,18)(19,23,24,25)(20,21,27,29)(22,31,32,33)(26,28,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612444);