Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36036 |
| 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 | |
5432 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);(3,4,8,7)(5,10,6,9)(11,12,15,14)(13,17,18,16)(19,20,24,22)(21,27,28,25)(23,26,31,30)(29,33,34,32);(3,5,8,6)(4,9,7,10)(11,13,15,18)(12,16,14,17)(21,28)(25,27)(29,34)(32,33);(3,6,8,5)(4,10,7,9)(11,12,15,14)(13,16,18,17)(19,21)(20,25)(22,27)(23,29)(24,28)(26,32)(30,33)(31,34);(3,4,8,7)(5,9,6,10)(19,22,24,20)(21,27,28,25)(23,26,31,30)(29,32,34,33);(3,7,8,4)(5,10,6,9)(11,14,15,12)(13,17,18,16)(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536714);