Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#1441 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C22×C42 |
| Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,8)(10,11)(17,18,19,20);(9,10,12,11)(13,14)(15,16);(6,8)(9,11,12,10)(14,16)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72298162, #72298300, #72298371, … |
| maximal normal subgroup of | #72298162, #72298300, #72298371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60939);