Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#28892 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,16)(14,15)(17,18)(19,20,22,25)(21,27,29,24)(23,26,28,32)(30,34,31,33);(3,5,6,10)(4,7,8,9)(19,21,28,31,22,29,23,30)(20,24,32,34,25,27,26,33);(12,15)(16,18)(19,21,22,29)(20,24,25,27)(23,31,28,30)(26,34,32,33);(4,8)(7,9)(11,13,14,17)(12,16,15,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367286);