Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#43821 |
| 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 | |
5452 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)(5,6)(7,9)(8,10)(11,12,15,18)(13,16,14,17)(19,20,24,26)(21,23,27,32)(22,25,31,34)(28,30,29,33);(3,5,8,7)(4,9,10,6)(11,13,15,14)(12,16,18,17)(21,27)(23,32)(28,29)(30,33);(11,12)(13,16)(14,17)(15,18)(19,21)(20,23)(22,29)(24,27)(25,33)(26,32)(28,31)(30,34);(3,6,8,9)(4,5,10,7)(19,22)(20,25)(21,28)(23,30)(24,31)(26,34)(27,29)(32,33);(3,7,8,5)(4,9,10,6)(11,14,15,13)(12,17,18,16)(19,23)(20,21)(22,30)(24,32)(25,28)(26,27)(29,34)(31,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7575284);