Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#32711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3134 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);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,20,24,23)(21,25,28,31)(22,26,30,32)(27,33,34,29);(3,5)(4,7)(6,10)(8,9)(11,13)(12,15)(14,17)(16,18)(19,21,24,28)(20,25,23,31)(22,29,30,33)(26,27,32,34);(3,5)(4,8)(6,10)(7,9)(12,16)(15,18)(19,22,24,30)(20,26,23,32)(21,27,28,34)(25,33,31,29);(5,10)(7,8)(13,17)(15,18)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538299);