Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
3598 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)(6,8)(9,10,11,12)(13,14)(15,19)(16,18)(17,20);(2,4)(13,15)(14,17,18,19)(16,20);(5,6,7,8)(13,14)(15,17,20,19)(16,18)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72260737, #72260761, #72260795, … |
| maximal normal subgroup of | #72260737, #72260761, #72260795, … |
| maximal quotient of | #72261112, #72280720, #72280758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31234);