Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#26211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2234 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,15)(16,18)(19,20,23,24)(21,25,28,32)(22,29,31,26)(27,30,34,33);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,21)(20,25)(22,30)(23,28)(24,32)(26,27)(29,34)(31,33);(11,13,14,18)(12,16,15,17)(19,22,23,31)(20,26,24,29)(21,27,28,34)(25,33,32,30);(4,8)(7,9)(12,15)(16,17)(19,20,23,24)(21,25,28,32)(22,26,31,29)(27,33,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349008);