Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,20,23,27)(21,25,29,24)(22,26,31,34)(28,33,30,32);(4,7)(5,10)(12,16)(13,18)(19,21,23,29)(20,24,27,25)(22,30,31,28)(26,33,34,32);(4,7)(8,9)(11,13)(12,14)(15,18)(16,17)(19,21)(20,25)(22,30)(23,29)(24,27)(26,32)(28,31)(33,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,22,23,31)(20,26,27,34)(21,28,29,30)(24,32,25,33);(3,5)(4,8)(6,10)(7,9)(11,14)(12,13)(15,17)(16,18)(19,20)(21,25)(22,26)(23,27)(24,29)(28,33)(30,32)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010718);