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×#31556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2188 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)(5,7)(6,9)(8,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,26)(21,24,29,33)(22,25,30,34)(27,31,28,32);(3,5)(4,6)(7,10)(8,9)(11,13,14,17)(12,15,16,18)(19,21)(20,24)(22,27)(23,29)(25,31)(26,33)(28,30)(32,34);(3,6,8,7)(4,9,10,5)(12,16)(15,18)(21,27)(22,30)(24,31)(25,34)(28,29)(32,33);(3,7,8,6)(4,9,10,5)(12,16)(15,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412265);