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×#17727 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3070 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)(5,14)(6,11)(7,12)(8,17)(9,13)(10,15)(16,18)(19,20,23,27,24,28,22,26)(21,29,30,25,32,33,31,34);(3,5,9,6)(4,10,13,18)(7,15,8,16)(11,17,14,12)(19,21)(20,25)(22,30)(23,31)(24,32)(26,33)(27,29)(28,34);(3,6,9,5)(4,11,13,14)(7,15,8,16)(10,17,18,12)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29);(3,7,9,8)(4,12,13,17)(5,15,6,16)(10,11,18,14)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251192);