Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| 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×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(9,10,16,19,18,22,23,12,17,20,13,24,14,11,15,21);(1,2,4,3)(5,6)(7,8)(9,11,17,22)(10,18,20,14)(12,23,21,15)(13,24,16,19);(1,3,4,2)(6,8)(9,12,13,20,18,24,15,11,17,21,16,10,14,19,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309661, #72309676, #72309695, … |
| maximal normal subgroup of | #72309661, #72309676, #72309695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56284);