Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53336 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4920 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);(3,4,6,7)(5,9,10,8)(11,12)(13,18)(14,17)(15,16)(21,27)(25,30)(28,33)(32,34);(4,7)(8,9)(12,16)(13,14)(20,24)(22,29)(25,30)(28,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,14)(12,17,16,18)(19,21)(20,25)(22,28)(23,27)(24,30)(26,32)(29,33)(31,34);(3,5,6,10)(4,8,7,9)(11,14,15,13)(12,18,16,17)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013072);