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×#32478 |
| 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 | |
1554 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);(3,4,6,7)(5,9,10,8)(11,12,15,17)(13,16,14,18)(19,20,22,25)(21,24,28,32)(23,26,30,33)(27,34,29,31);(4,7)(5,10)(11,13,15,14)(12,16,17,18)(19,21)(20,24)(22,28)(23,27)(25,32)(26,34)(29,30)(31,33);(11,14,15,13)(12,16,17,18)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34);(3,5,6,10)(4,8,7,9)(12,17)(16,18)(20,25)(24,32)(26,33)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417032);