Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19086 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,12)(6,7)(11,14)(13,15)(16,18)(19,20)(21,27)(22,30)(23,25)(24,29)(26,32)(28,34)(31,33);(3,4)(5,10)(6,14)(7,9)(8,11)(12,18)(13,17)(15,16)(19,21)(20,24)(22,31)(23,28)(25,33)(26,34)(27,30)(29,32);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,17)(14,18)(22,32)(26,30)(27,34)(29,31);(4,11)(9,14)(10,17)(16,18)(20,25)(21,28)(26,30)(29,31)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258900);