Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C8 |
| direct product | C4×#1720 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(5,6,10,7)(8,9,12,11)(13,14,17,20)(15,21,23,28)(16,25,22,19)(18,26,27,24);(5,7,9,12,10,6,11,8)(13,15,22,26,17,23,16,24)(14,18,25,28,20,27,19,21);(5,8,9,7,10,12,11,6)(15,23)(18,27)(21,28)(24,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120305);