Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8321 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #379 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,8)(6,13)(9,17)(10,14)(11,15)(16,18)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,4)(5,9)(6,14)(7,10)(8,13)(11,16)(12,18)(15,17)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(3,5,7,12)(4,9,10,18)(6,15,13,11)(8,16,14,17)(21,27)(24,30)(28,33)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92130);