Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17839 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1682 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)(5,12)(6,10)(7,17)(8,11)(9,15)(13,14)(16,18)(19,20,23,27,24,28,22,26)(21,29,32,33,30,34,31,25);(3,5,7,15,8,16,6,14)(4,9,10,12,11,13,17,18)(19,21,22,31,24,30,23,32)(20,25,27,29,28,33,26,34);(3,4)(5,13)(6,17)(7,10)(8,11)(9,16)(12,14)(15,18)(21,30)(25,33)(29,34)(31,32);(3,6,8,7)(4,10,11,17)(5,14,16,15)(9,12,13,18)(19,22,24,23)(20,26,28,27)(21,31,30,32)(25,34,33,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251580);