Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38395 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4974 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);(13,17)(15,18)(19,20,24,25)(21,27,30,26)(22,31,32,23)(28,29,34,33);(4,7)(8,10)(11,12,14,16)(13,15,17,18)(20,25)(23,31)(26,27)(29,33);(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21,24,30)(20,26,25,27)(22,28,32,34)(23,29,31,33);(3,5,6,9)(4,8,7,10)(19,22)(20,23)(21,28)(24,32)(25,31)(26,29)(27,33)(30,34);(3,4)(5,8)(6,7)(9,10)(19,23)(20,22)(21,29)(24,31)(25,32)(26,28)(27,34)(30,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543010);