Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#2821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2);(3,4);(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,28)(31,32);(6,8)(11,15)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(9,11)(10,13)(12,15)(14,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540597);