Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17725 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2886 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);(4,8)(6,13)(9,15)(10,14)(12,17)(16,18)(19,20,23,27,24,28,22,26)(21,29,30,25,32,33,31,34);(3,4,7,10)(5,9,11,16)(6,14,13,8)(12,18,17,15)(19,21)(20,25)(22,30)(23,31)(24,32)(26,33)(27,29)(28,34);(3,5,7,11)(4,9,10,16)(6,12,13,17)(8,15,14,18)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29);(5,11)(9,16)(12,17)(15,18)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251190);