Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36258 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(14,15)(16,17)(19,20);(5,6,8,9)(7,10,11,12)(13,14,16,15,18,20,17,19);(2,4)(6,9)(7,11)(13,15,16,20,18,19,17,14);(5,7)(6,10)(8,11)(9,12);(6,9)(7,11)Polycyclic presentation:
2^9|b^2=fhi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283113, #72283186, #72283302 |
| maximal normal subgroup of | #72283113, #72283186, #72283302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505477);