Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,14)(9,15)(10,17)(11,18)(12,16)(19,20,23,27)(21,24,28,32)(22,25,30,26)(29,33,34,31);(3,4,7,9)(5,11,13,17)(6,15,14,8)(10,16,18,12)(19,21)(20,24)(22,29)(23,28)(25,33)(26,31)(27,32)(30,34);(4,9)(8,15)(10,18)(11,17)(20,25)(21,28)(22,30)(24,31)(26,27)(32,33);(3,5,7,13)(4,10,9,18)(6,16,14,12)(8,17,15,11)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266731);