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×#33370 |
| 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×D4 ◁ C22×2−1+4 ◁ 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);(3,4,7,12,8,13,20,27)(5,10,16,25,14,23,29,33)(6,11,18,26,17,9,19,22)(15,24,30,34,28,21,31,32);(4,9)(6,17)(10,21)(11,13)(12,26)(15,28)(18,19)(22,27)(23,24)(25,34)(30,31)(32,33);(3,5)(4,10)(6,15)(7,16)(8,14)(9,21)(11,24)(12,25)(13,23)(17,28)(18,30)(19,31)(20,29)(22,32)(26,34)(27,33);(5,14)(10,23)(15,28)(16,29)(21,24)(25,33)(30,31)(32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529674);