Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,12,16,14,19,13,18)(11,20,22,24,23,15,21,17);(10,15,19,20)(11,21)(12,13)(16,24,18,17)(22,23);(2,4)(10,16,19,18)(15,17,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214428, #72214436, #72215353, … |
| maximal normal subgroup of | #72214428, #72214436, #72215353, … |
| maximal quotient of | #72214231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56654);