Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27621 |
| 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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(5,10)(6,8)(7,9)(11,12,14,16)(13,15,17,18)(19,20,25,29)(21,26,30,27)(22,31,32,34)(23,28,24,33);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,26)(22,24)(23,32)(25,30)(27,29)(28,34)(31,33);(3,6,7,5)(4,8,9,10)(13,17)(15,18)(19,22,25,32)(20,27,29,26)(21,23,30,24)(28,34,33,31);(4,9)(8,10)(19,23,25,24)(20,28,29,33)(21,22,30,32)(26,31,27,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366098);