Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 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);(3,4)(5,7)(6,8)(9,10)(11,12,15,14)(13,17,18,16)(19,20,23,26)(21,24,28,32)(22,25,29,33)(27,31,30,34);(3,5)(4,7)(6,9)(8,10)(11,13,15,18)(12,16,14,17)(21,27)(22,29)(24,31)(25,33)(28,30)(32,34);(4,8)(7,10)(12,14)(16,17)(19,21,23,28)(20,24,26,32)(22,30,29,27)(25,34,33,31);(5,9)(7,10)(11,14,15,12)(13,17,18,16)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538363);