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×#50010 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5560 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);(11,12)(13,14)(15,17)(16,18)(19,20,24,27)(21,22,29,25)(23,31,32,26)(28,33,34,30);(4,7)(5,9)(13,15)(14,17)(19,21,24,29)(20,25,27,22)(23,28,32,34)(26,33,31,30);(3,4,6,7)(5,8,9,10)(11,13,16,15)(12,17,18,14)(19,22)(20,21)(23,30)(24,25)(26,28)(27,29)(31,34)(32,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,17)(12,13,18,15)(19,21)(20,22)(23,28)(24,29)(25,27)(26,30)(31,33)(32,34);(3,5)(4,8)(6,9)(7,10)(11,15,16,13)(12,17,18,14)(19,23,24,32)(20,26,27,31)(21,28,29,34)(22,30,25,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612442);