Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C43 |
| direct product | C4×#1436 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C2×C43 |
| Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4670 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)(9,10)(11,12)(13,14,16,15);(6,8)(17,18,19,20);(10,12)(13,15,16,14)224 conjugacy classes — view the full character table.
| maximal subgroup of | #72298489, #72298490, C4×#15310, … |
| maximal normal subgroup of | #72298489, #72298490, C4×#15310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60909);