Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | |
572 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)(9,10,13,18,14,21,12,19,15,20,11,17,23,24,22,16);(2,4)(5,6)(7,8)(10,16,20,19)(11,22)(12,13)(14,23)(17,24,18,21);(2,4)(6,8)(10,17,20,18)(11,22)(12,13)(14,23)(16,21,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214409, #72214416, #72216348, … |
| maximal normal subgroup of | #72214409, #72214416, #72216348, … |
| maximal quotient of | #72214231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56545);