Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15,16,13)(12,14)(18,22,23,24);(2,4)(9,11,12,16)(10,13,14,15)(17,18)(19,24)(20,22)(21,23);(6,8)(13,15)(18,22)(19,20)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72260923, #72260983, #72260985 |
| maximal normal subgroup of | #72260923, #72260983, #72260985 |
| maximal quotient of | #72261280, #72261287 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22094);