Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52943 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5978 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)(5,9)(6,8)(7,10)(12,17)(13,15)(22,30)(26,31)(28,34)(32,33);(4,6)(9,10)(11,12,16,17)(13,14,15,18)(20,24)(25,27)(26,31)(32,33);(3,5,8,7)(4,9,6,10)(11,13,16,15)(12,18,17,14)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(3,6,8,4)(5,9,7,10)(11,14,16,18)(12,13,17,15)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,7,8,5)(4,9,6,10)(11,15,16,13)(12,18,17,14)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999928);