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 | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14,18,15)(16,19,17,20);(2,4)(9,11)(14,19)(15,20)(16,17);(5,7)(6,9)(8,12)(10,11)(13,15,18,14)(16,20,17,19)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=ghi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240315, #72240394, #72240397 |
| maximal normal subgroup of | #72240315, #72240394, #72240397 |
| maximal quotient of | #72261287 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22529);