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×#50882 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5136 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);(3,4,8,10)(5,7,6,9)(11,12,15,13)(14,17,18,16)(19,20,24,27)(21,25,28,33)(22,23,26,32)(29,34,31,30);(3,5,8,6)(4,9,10,7)(11,13,15,12)(14,17,18,16)(21,28)(22,26)(23,32)(25,33);(4,10)(5,6)(11,14)(12,16)(13,17)(15,18)(19,21)(20,25)(22,31)(23,30)(24,28)(26,29)(27,33)(32,34);(3,6,8,5)(4,9,10,7)(19,22,24,26)(20,23,27,32)(21,29,28,31)(25,34,33,30);(3,7,8,9)(4,5,10,6)(19,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,34)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613864);