Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#2966 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,19,23,26)(15,18,21,25,22,24,27,28);(6,9)(10,12);(13,15,16,21,17,22,23,27)(14,18,20,25,19,24,26,28);(5,7,8,11)(6,10,9,12)(14,19)(18,24)(20,26)(25,28)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277341);