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×#50387 |
| 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 | |
5120 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)(5,9)(6,8)(7,10)(13,17)(16,18)(19,20,24,25)(21,27,28,22)(23,31,32,26)(29,30,34,33);(3,4,6,8)(5,7,10,9)(11,12,15,14)(13,16,17,18)(20,25)(21,28)(23,32)(30,33);(11,13)(12,16)(14,18)(15,17)(19,21,24,28)(20,22,25,27)(23,29,32,34)(26,30,31,33);(3,4)(5,7)(6,8)(9,10)(19,22)(20,21)(23,30)(24,27)(25,28)(26,29)(31,34)(32,33);(3,5)(4,7)(6,10)(8,9)(11,14,15,12)(13,18,17,16)(19,23,24,32)(20,26,25,31)(21,29,28,34)(22,30,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612949);