Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1240 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)(7,9)(8,10)(11,12)(13,14,18,20)(15,16,19,17);(5,6)(7,11,12,9)(8,10)(13,15,16,20,18,19,17,14);(5,7)(6,9)(8,12)(10,11)(14,19)(15,20)(16,17)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fhi,d^c=di,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263995, #72264012, #72264048, … |
| maximal normal subgroup of | #72263995, #72264012, #72264048, … |
| maximal quotient of | #72280223, #72281354 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25509);