Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#31971 |
| 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 | |
1684 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);(3,4,8,9)(5,7,10,6)(11,12,14,16)(13,15,17,18)(19,20,23,24)(21,27,29,25)(22,30,32,26)(28,33,34,31);(3,5,8,10)(4,6,9,7)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,31)(23,29)(24,27)(26,28)(30,34)(32,33);(3,6,8,7)(4,5,9,10)(12,16)(15,18)(19,22,23,32)(20,26,24,30)(21,28,29,34)(25,33,27,31);(3,7,8,6)(4,5,9,10)(12,16)(15,18)(21,29)(25,27)(28,34)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413991);