Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2023 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,17,23,16)(7,18,20,33,21,30,24,13)(8,22,25,32,19,12,29,31)(9,26,27,14,28,34,36,35);(1,2)(3,4)(5,7,9,8,23,24,36,29,11,21,28,19,10,20,27,25)(6,12,14,30,15,31,34,13,17,22,35,18,16,32,26,33);(2,4)(5,8,9,20,23,29,36,7,11,19,28,24,10,25,27,21)(6,13,14,12,15,18,34,31,17,33,35,22,16,30,26,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250594, #72250609, #72250618, … |
| maximal normal subgroup of | #72250594, #72250609, #72250618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56183);