Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
4670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(13,14)(15,17,20,18)(16,19);(2,4)(9,10)(11,12)(13,15)(14,17,19,18)(16,20);(6,8)(10,12)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72219232, #72219233, #72225643, … |
| maximal normal subgroup of | #72219232, #72219233, #72225643, … |
| maximal quotient of | #72219228, #72219230, #72250028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30786);