Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×Q8 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,14,10,15,6,12)(5,16,17,32,19,27,24,34)(7,13,21,31,8,22,20,30)(11,25,28,23,29,33,26,18);(3,5)(4,11)(6,17)(7,18)(8,23)(9,24)(10,19)(12,26)(13,27)(14,28)(15,29)(16,22)(20,25)(21,33)(30,32)(31,34);(3,6,10,9)(4,12,15,14)(5,17,19,24)(7,21,8,20)(11,26,29,28)(13,31,22,30)(16,32,27,34)(18,33,23,25);(3,7,10,8)(4,13,15,22)(5,18,19,23)(6,20,9,21)(11,27,29,16)(12,30,14,31)(17,25,24,33)(26,32,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529676);