Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,15,12,18,29,35,10,17,9,16,27,30,28,36)(7,19,21,32,23,33,25,31,22,34,20,14,8,24,26,13);(1,2)(3,4)(5,7,12,8,10,22,27,23)(6,13,30,32,17,31,18,14)(9,26,28,20,11,25,29,21)(15,33,36,34,16,24,35,19);(2,4)(5,8,12,22,10,23,27,7)(6,14,30,13,17,32,18,31)(9,21,28,26,11,20,29,25)(15,34,36,24,16,19,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56550);