Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#31517 |
| 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 | |
2026 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);(4,7)(5,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,24)(21,27,29,25)(22,26,31,32)(28,30,34,33);(3,4,6,7)(5,10,9,8)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,30)(23,29)(24,27)(26,28)(31,33)(32,34);(12,16)(15,18)(19,22,23,31)(20,26,24,32)(21,28,29,34)(25,33,27,30);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(21,29)(25,27)(28,34)(30,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412104);