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 | C2×C4 | #476 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1920 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,13,10,15,29,26,11,16,30,8,24,14,35,27)(7,17,21,18,22,34,36,12,23,32,19,31,20,25,28,33);(1,2,4,3)(5,7,10,20,11,23,24,22)(6,12,14,31,16,33,15,18)(8,25,26,17,13,34,27,32)(9,28,29,19,30,36,35,21);(1,3,4,2)(5,8,9,14,10,27,29,6,11,13,30,15,24,26,35,16)(7,18,19,25,22,12,28,17,23,31,21,34,20,33,36,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=ghi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56245);