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 | #1919 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,32,23,12,18,9,15,24,19,29,35)(7,8,20,25,22,26,28,31,14,27,21,36,33,34,30,13);(2,4)(5,7)(6,13,18,31)(8,23,27,35)(9,28)(10,30)(11,33)(12,14)(15,26,16,34)(17,36,19,25)(20,29)(21,32)(22,24);(2,4)(5,8,24,26,12,27,11,34)(6,14,17,22,18,7,19,33)(9,13,32,25,10,31,29,36)(15,28,35,20,16,30,23,21)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56589);